 |
 | Home | Contact Us | FAQ | Search & Site Map | Link to Us |
 | Sign In | Join | Other 45 Sites in Network |
Math Forum / Mathematics / Mathematical Logic / January 2008Torkel Franzen on truth
In "Gödel's theorem" Torkel Franzen disputes that the theorem indicates that the human mind surpasses any computer. >> ... the mistaken idea that "Gödel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulas which cannot be proved in the system, but which we can see to be true." The theorem states no such thing. As has been emphasized, in general we simply have no idea whether or not the Gödel sentence of a system is true, even in those cases when it is in fact true. What we know is that the Gödel sentence is true if and only if the system is consistent, and that much is provable in the system itself. << p. 55 >> ... there is no doubt whatever about the consistency of any of the formal systems we use in mathematics. << p. 105 >> If the axioms of ZFC are manifestly true, they are obviously consistent. << p. 105 I am not sure that I understand what Franzen is saying. Is he saying that a) We are absolutely certain about the truths of PA, even those PA cannot prove b) The consistency of PA can be proven in ZFC c) Therefore we can write a computer program emulating ZFC that can generate the truths of PA d) We are absolutely certain about the truths of ZFC, even those ZFC cannot prove e) There is a theory X in which we can prove the consistency of ZFC f) Therefore we can write a computer program emulating X that can generate the truths of ZFC g) We are not certain about the truths of X ?? > In "Gödel's theorem" Torkel Franzen disputes that the theorem > indicates that the human mind surpasses any computer. [quoted text clipped - 4 lines] > > I am not sure that I understand what Franzen is saying. The first quote you give, I take it is entirely clear (and correct!). Your second quote is misleading: What TF in fact wrote was "Nothing in Gödel's theorem in any way contradicts the view that there is no doubt whatever about the consistency of any of the formal systems we use in mathematics." TF isn't there endorsing the view (as your truncated quotation suggests), he is just pointing out that Godel's theorem doesn't refute it -- a point evidently consistent with the first quote. The third quote you give starts with an emphasized "If" in TF. It is a triviality (any set of truths is consistent!). He is not, at least in those quotations, saying any of (a) to (g). > > In "Gödel's theorem" Torkel Franzen disputes that the theorem > > indicates that the human mind surpasses any computer. [quoted text clipped - 6 lines] > > The first quote you give, I take it is entirely clear (and correct!). This I don`t understand: "As has been emphasized, in general we simply have no idea whether or not the Gödel sentence of a system is true, even in those cases when it is in fact true." I have thought: If we assume the consistency of PA we can proof in PA + con(PA) that the gödel sentence of PA being true but not-provable (thus it follows from this that also the con(PA) is not provable from the axioms of PA). And certainly we have an least an informally "idea" that PA is consistent. > > > In "Gödel's theorem" Torkel Franzen disputes that the theorem > > > indicates that the human mind surpasses any computer. [quoted text clipped - 17 lines] > the axioms of PA). And certainly we have an least an informally "idea" > that PA is consistent. Oh wait. OK. Now I understand? The point is in GENERAL we don`t have an idea if the gödel sentence is true, like in New Foundations? > > > > In "Gödel's theorem" Torkel Franzen disputes that the theorem > > > > indicates that the human mind surpasses any computer. [quoted text clipped - 20 lines] > Oh wait. OK. Now I understand? The point is in GENERAL we don`t have > an idea if the gödel sentence is true, like in New Foundations? Yes, I think that's what TF was after: we'll in general not know whether T's standard Gödel's sentence is true because we'll not know whether T is consistent. (Of course, we are usually interested in theories T which we have pretty good reason to think are consistent, and we are usually not interested in the other cases! TF is just reminding us of the general situation.) > > > > > In "Gödel's theorem" Torkel Franzen disputes that the theorem > > > > > indicates that the human mind surpasses any computer. [quoted text clipped - 27 lines] > and we are usually not interested in the other cases! TF is just > reminding us of the general situation.) Do you mean that PA is PROBABLY consistent? > > > > > > In "Gödel's theorem" Torkel Franzen disputes that the theorem > > > > > > indicates that the human mind surpasses any computer. [quoted text clipped - 29 lines] > > Do you mean that PA is PROBABLY consistent? Read "pretty good reason" to mean "at least pretty good reason, maybe conclusive reason". As it happens I think there are conclusive reasons to believe PA consistent. > Read "pretty good reason" to mean "at least pretty good reason, maybe > conclusive reason". As it happens I think there are conclusive reasons > to believe PA consistent. Yes, PA is obviously consistent. -- Aatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus >> Read "pretty good reason" to mean "at least pretty good reason, maybe >> conclusive reason". As it happens I think there are conclusive reasons >> to believe PA consistent. > > Yes, PA is obviously consistent. So what?  SignatureCheers, Herman Jurjus > aatu.koskensi...@xortec.fi wrote: > > Yes, PA is obviously consistent. > > So what? So N is a model of it, so G is true, so a whole buncha stuff. Pick one. I don't think AK will care as much as Newberry will. >> Yes, PA is obviously consistent. > > So what? Nothing much. Why? SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus On Nov 9, 7:17 am, aatu.koskensi...@xortec.fi wrote: > Yes, PA is obviously consistent. Come on. The Gentzen proof is not obviously understandable. N doesn't obviously exist at all. > Come on. > The Gentzen proof is not obviously understandable. > N doesn't obviously exist at all. Gentzen's proof and the existence of the set of natural numbers don't really have much to do with obviousness of consistency of PA. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus On Nov 12, 6:43 am, Aatu Koskensilta <aatu.koskensi...@xortec.fi> wrote: > Gentzen's proof and the existence of the set of natural numbers don't really > have much to do with obviousness of consistency of PA. Are you saying that PA continues to remain obviously consistent even in the context where it is insisted that N is a proper classm and just plain can't be a set? Given that the model existence theorem is a theorem of SET theory, that is going to be a little complicated. There is a context in which we can PROVE that a theory is consistent if and only if it has a model. If N doesn't exist then it isn't a model. So is it always obvious how to prove that some other model exists? Or are you insisting that some context-for-debating-consistency-in- which- models-in-general-are-not-relevant has *higher*, *prior* claim or status than THE USUAL context, in which models of consistent theories MUST exist? > Are you saying that PA continues to remain obviously consistent > even in the context where it is insisted that N is a proper classm and > just plain can't be a set? Sure. No sets or classes need be involved in the observation that PA is obviously consistent. Of course, if we do not assume that infinite sets exist, it can no longer be established that a theory is consistent just in case it has a model on the standard definition of "model". SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus On Nov 9, 4:17 am, aatu.koskensi...@xortec.fi wrote: > > Read "pretty good reason" to mean "at least pretty good reason, maybe > > conclusive reason". As it happens I think there are conclusive reasons > > to believe PA consistent. > > Yes, PA is obviously consistent. Con(T) --> T(G) --> G We have just proven G. Now there are two possibilities a) The human mind surpasses any machine b) The human mind does not surpass a machine TF is in favor of b. In the first case the human mind can see the axioms of PA as manifestly true, which a machine cannot. If b is the case then I wonder how we can construct a machine that can generate all the truth of PA. > b) The human mind does not surpass a machine > [quoted text clipped - 3 lines] > wonder how we can construct a machine that can generate all the truth > of PA. Why should any machine be able to generate all the truths of (the language of) PA? After all, we can't do that either. > > b) The human mind does not surpass a machine > [quoted text clipped - 6 lines] > Why should any machine be able to generate all the truths of (the > language of) PA? After all, we can't do that either. How can we construct a machine that can generate all the truth of PA that we can? > How can we construct a machine that can generate all the truth of PA > that we can? Well, who knows which truths *those* are? > > > b) The human mind does not surpass a machine > [quoted text clipped - 9 lines] > How can we construct a machine that can generate all the truth of PA > that we can? Good question. In 'Inexhaustibility' TF poses the following question: It seems that whenever human logico-mathematical reason (HLMR) sees as evident a set of axioms, it also sees as evident the proposition that those axioms are consistent (which is a kind of reflection principle). But, if HLMR is consistent and sufficiently rich, that proposition does not always follow from those axioms (by G?del's second theorem). So, if there is an initial and sufficiently rich set of logico- mathematical truths that must be included in HLMR and HMLR is closed under that kind of reflection principle, there is no algorithm representing human logico-mathematical reason. As I interpret TF, he denies the conclusion by alleging 1. It might happen that there is no such thing as a definite HLMR. 2. Even if HLMR exists, human finiteness precludes the possibility that it is closed under that reflection principle: humans will hesitate as things grow increasingly involved. TF's position (very akin indeed to Hofstadter's) seems questionable to me because it fails to recognize the existence of an ideal legality in human reason, that is different from what humans can actually perform, and that he, TF, is implicitly invoking while reasoning. Nevertheless, I think TF's arguments show clearly why Lucas's and Penrose's arguments fail. They both are assuming implicitly that: A. HLMR is a definite object B. HLMR is closed under some reflection principle(s). Clearly, A and B does not follow from G?del's theorem. Regards > > > > b) The human mind does not surpass a machine > [quoted text clipped - 36 lines] > human reason, that is different from what humans can actually perform, > and that he, TF, is implicitly invoking while reasoning. This is the part I did not understand. And this is how I interpreted it: a) We are absolutely certain about the truths of PA, even those PA cannot prove b) The consistency of PA can be proven in ZFC c) Therefore we can write a computer program emulating ZFC that can generate the truths of PA d) We are absolutely certain about the truths of ZFC, even those ZFC cannot prove e) There is a theory X in which we can prove the consistency of ZFC f) Therefore we can write a computer program emulating X that can generate the truths of ZFC g) We are not certain about the truths of X Maybe I read him wrong? > Nevertheless, I think TF's arguments show clearly why Lucas's and > Penrose's arguments fail. They both are assuming implicitly that: [quoted text clipped - 7 lines] > > - Show quoted text - > a) We are absolutely certain about the truths of PA, even those PA > cannot prove Eh??? Who on earth is committing themselves to that silly claim??? Suppose L_1 is the language of first-order arithmetic, then there are various classes of truths of L_1. 1. There are the truths for which we can actually give a proof in first-order PA. 2. There are truths (like the arithmetization of Goodstein's theorem, or like Con(PA)) which are provably not provable in PA, but for which we have proofs in other, richer, theories -- like suitable fragments of set theory. 3. There are other truths which it is not known whether or not they are provable in first-order PA, though we have do proofs in other richer theories. 4. And no doubt there are other truths for which we have no kind of proof (as yet: or may be there could be no humanly surveyable proof). Plainly these different sorts of truths have different epistemic status! We might be certain of the truth of the propositions in the first three classes, given we accept the relevant proofs. But we certainly not certain about the truths in the fourth class (even if we strongly suspect that some propositions like Goldbach's conjecture do indeed fall in the class of truth-but-not-yet proved). > > a) We are absolutely certain about the truths of PA, even those PA > > cannot prove [quoted text clipped - 10 lines] > for which we have proofs in other, richer, theories -- like suitable > fragments of set theory. There is also 5. the truths of which we are absolutely certain although they are provably unprovable in PA, like Con(T). The certainty comes from seeing that the axioms of PA as manifestly true. Perhaps 5 is identical with 2. How do we know that these richer theories, such as a suitable fragment of ZFC, are consistent? > 3. There are other truths which it is not known whether or not they > are provable in first-order PA, though we have do proofs in other [quoted text clipped - 8 lines] > strongly suspect that some propositions like Goldbach's conjecture do > indeed fall in the class of truth-but-not-yet proved). > > > > > b) The human mind does not surpass a machine > [quoted text clipped - 69 lines] > > - Show quoted text - I interpret that your a), b), c),... are just an example (for actually they are not true). If so, what you say is similar to what I think TF said. TF suggests that we would eventually get to such a convoluted theory that we would hesitate in applying the reflection principles in order to get new truths. We do believe: a') The axioms of PA b') a')+ Con(a') c') b') + Con(b') etc. TF says that at some too high level things would get so difficult that we would stagger or would be simply unable to go on. What I argue is that though this is in fact so, the argument fails to distinguish the logico-mathematical legality enclosed in human reason from what humans can effectively accomplish., i.e. what is logically possible for humans from what is physically possible for them. The necessity of reducing logical impossibility to physical impossibility is one of the weak points of AI. Regards > > > > b) The human mind does not surpass a machine > [quoted text clipped - 44 lines] > > Clearly, A and B does not follow from G?del's theorem. Where do Lucas and Penrose asume A and B. The issue is that we can conclude with certainty that G is true. (PA is consistent because the axioms are manifestly true.) Thus far no one explained how we could construct a machine that would do the same. On Nov 9, 4:17 am, aatu.koskensi...@xortec.fi wrote: > > Read "pretty good reason" to mean "at least pretty good reason, maybe > > conclusive reason". As it happens I think there are conclusive reasons > > to believe PA consistent. > > Yes, PA is obviously consistent. OK, how do we reconcile it with this? >> ... the mistaken idea that "Gödel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulas which cannot be proved in the system, but which we can see to be true." The theorem states no such thing. As has been emphasized, in general we simply have no idea whether or not the Gödel sentence of a system is true, even in those cases when it is in fact true. What we know is that the Gödel sentence is true if and only if the system is consistent, and that much is provable in the system itself. << p. 55 > On Nov 9, 4:17 am, aatu.koskensi...@xortec.fi wrote:> On 9 Oct, 10:20, Peter_Smith wrote: > [quoted text clipped - 7 lines] > > >> ... the mistaken idea that "G?del's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulas which cannot be proved in the system, but which we can see to be true." The theorem states no such thing. As has been emphasized, in general we simply have no idea whether or not the G?del sentence of a system is true, even in those cases when it is in fact true. What we know is that the G?del sentence is true if and only if the system is consistent, and that much is provable in the system itself. << p. 55 There is no conflict at all between what I said (something TF held too), and that latter quote. To hold that PA is clearly consistent is quite compatible with holding that, with some arbitrarily thrown- together extension of Q, we won't in the general case know whether it is consistent, and hence won't know whether its canonical G?del sentence is true. > > On Nov 9, 4:17 am, aatu.koskensi...@xortec.fi wrote:> On 9 Oct, 10:20, Peter_Smith wrote: > [quoted text clipped - 14 lines] > is consistent, and hence won't know whether its canonical G?del > sentence is true. So let's confine ourselves to PA for now. We can prove that it is consistent, that is we have proven G. How did we manage to do that without running into a contradiction? We did not simply add Con(T) as another axiom, we proved it. I suppose we proved it in some metatheory M. How do we know that M is consistent? > > > On Nov 9, 4:17 am, aatu.koskensi...@xortec.fi wrote:> On 9 Oct, 10:20, Peter_Smith wrote: > [quoted text clipped - 20 lines] > another axiom, we proved it. I suppose we proved it in some metatheory > M. How do we know that M is consistent? Depends what M is: it if is a suitable set theory, by getting your head around the idea of the structure of the iterative hierarchy. (I know this might sound odd coming from someone whose day-job is as a philosopher, but frankly, I do find "how we know?" questions are as entirely boring applied to maths as applied to claims about medium- sized dry goods. Scepticism either way is just uninteresting.) > > How do we know that M is consistent? > [quoted text clipped - 5 lines] > entirely boring applied to maths as applied to claims about medium- > sized dry goods. Scepticism either way is just uninteresting.) Of course it's uninteresting to you. As a rule people like the way that they go and are not interested in reflecting on why they go that way. At the least, there's no profit in it. Aunt Bessie goes to the supermarket every Thursday, and she does not take kindly any suggestions that it might be possible to go on Friday. > > > How do we know that M is consistent? > [quoted text clipped - 9 lines] > that they go and are not interested in reflecting on why they go that > way. I'm interested in reflecting ... when given good reason to do so. > > > > How do we know that M is consistent? > [quoted text clipped - 11 lines] > > I'm interested in reflecting ... when given good reason to do so. Ah, well "good reason" is a term which, because of its subjectivity, does not advance matters at all. Aunt Bessie has good reason to go to the supermarket every Thursday; she has always done so. > > > > > How do we know that M is consistent? > [quoted text clipped - 15 lines] > does not advance matters at all. Aunt Bessie has good reason to go > to the supermarket every Thursday; she has always done so. Why is "good reason" subjective? And as for what Aunt Bessie has to do with the question whether we have good reason to doubt, e.g., the truth of PA, I'm completely stumped! > Why is "good reason" subjective? What is "good reason" to you may not be "good reason" to someone else. For instance, a hard-core theist would hold that there is not "good reason" to discuss the existence of God. > And as for what Aunt Bessie has to do > with the question whether we have good reason to doubt, e.g., the > truth of PA, I'm completely stumped! Well, Aunt Bessie has good reason to go shopping every Thursday; other people don't. Yet you ask (I presume with a straight face) why "good reason" is subjective or what Aunt Bessie has to do with it. > > Why is "good reason" subjective? > > What is "good reason" to you may not be "good reason" to someone > else. For instance, a hard-core theist would hold that there is not > "good reason" to discuss the existence of God. Bad argument. The hard core theist might hold that, but that doesn't obviously ential that they are right to do so. > > And as for what Aunt Bessie has to do > > with the question whether we have good reason to doubt, e.g., the [quoted text clipped - 3 lines] > people don't. Yet you ask (I presume with a straight face) why "good > reason" is subjective or what Aunt Bessie has to do with it. Bad argument. The fact that A has a good reason to do castle, and B has a good reason to not to do castle doesn't make either reason "subjective". A and B's situation in the game may be different, and it could -- for all that has been said -- be an objective matter that someone in A's position has a good reason to castle and someone in B's situation has a good reason not to castle. Mutatis mutandis for Bessie. > Bad argument. The fact that A has a good reason to do castle, and B > has a good reason to not to do castle doesn't make either reason [quoted text clipped - 3 lines] > situation has a good reason not to castle. Mutatis mutandis for > Bessie. Apologies -- that's very careless editing! "to do castle" should read, of course "to castle". > > > Why is "good reason" subjective? > [quoted text clipped - 4 lines] > Bad argument. The hard core theist might hold that, but that doesn't > obviously ential that they are right to do so. Yes, obviously you can objectify 'good reason' so that it just means "good reason to Peter Smith." Then the hard-core theist is of course wrong; his is not a good reason. Yet, what he means by the words 'good reason' is different from what you mean by the words 'good reason.' And in the end, both your words do the same work, in that he is not willing to reflect unless there is "good reason" just as you are not. > > > And as for what Aunt Bessie has to do > > > with the question whether we have good reason to doubt, e.g., the [quoted text clipped - 5 lines] > > Bad argument. Where was the argument? > The fact that A has a good reason to do castle, and B > has a good reason to not to do castle doesn't make either reason [quoted text clipped - 4 lines] > situation has a good reason not to castle. Mutatis mutandis for > Bessie. Not at all. I explained to you that Aunt Bessie's only reason for going shopping on Thursday was that she had always done so. She might take that as a good reason, but her grandson, who wants to take her to the museum and can only take Thursday off work, would not. (And I would presume you would not either, even though your epistemological views apparently tend it that direction.) > > > > Why is "good reason" subjective? > [quoted text clipped - 7 lines] > Yes, obviously you can objectify 'good reason' so that it just means > "good reason to Peter Smith." Sigh. I was of course doing no such thing. > What is "good reason" to you may not be "good reason" to someone > else. For instance, a hard-core theist would hold that there is not > "good reason" to discuss the existence of God. An indifferent atheist might also well find discussing the existence of God somewhat pointless. Regardless of the question of whether "good reason" is or is not subjective, it remains a rather trivial platitude that people will in fact be interested in subjecting this or that to scrutiny, reflection, doubt, only if presented some incentive to, a "good reason" in a more mundane sense. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus On Nov 12, 12:56 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi> wrote: > > What is "good reason" to you may not be "good reason" to someone > > else. For instance, a hard-core theist would hold that there is not [quoted text clipped - 6 lines] > doubt, only if presented some incentive to, a "good reason" in a more > mundane sense. Somebody, who was in conversation with PS about this subject, asked him a question, "How do you know?" That would seem to be incentive enough to provide at least a modicum of scrutiny or reflection. > Somebody, who was in conversation with PS about this subject, asked > him a question, "How do you know?" That would seem to be incentive > enough to provide at least a modicum of scrutiny or reflection. Surely you know the grounds on which we -- Peter, me, Torkel, and so on -- find PA's consistency obvious by now. On the conception that the naturals are obtained from 0 by repeatedly applying the "add-one"-operation the principle of induction Whenever P is a determinate mathematical property of naturals, if 0 has P, and whenever n has P, n+1 also has P, all naturals have P is manifestly true, as is the principle of definition by primitive recursion, that properties definable by primitive recursion are determinate and well-defined in the relevant sense. Combining this observation with the determinateness of properties expressible in the first order language of arithmetic, that is, those obtainable from the primitive recursive properties by means of the usual logical operations, leads immediately to the conclusion that the axioms of PA are all manifestly true, and hence no contradiction follows from them, by the soundness of the rules of inference of first order logic. Now, if someone finds this explanation incomprehensible even after elaborations, illustrations, gentle persuasion, practice, and so on, I'm stumped. There's simply nothing I can do but shrug. Of course, people might be interested in e.g. what can and cannot be proved without appeal to the totality of the successor function, full induction, etc. for perfectly sensible reasons -- often we obtain mathematical information beyond than just that P is true if we know that P is not only true but also provable from these or those (weak) principles -- but connecting such interests to rather elusive and incomprehensible doubts is pointless. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus On Nov 12, 6:59 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi> wrote: > > Somebody, who was in conversation with PS about this subject, asked > > him a question, "How do you know?" That would seem to be incentive [quoted text clipped - 27 lines] > from these or those (weak) principles -- but connecting such interests to > rather elusive and incomprehensible doubts is pointless. "Now, if someone finds this explanation incomprehensible..." Beware when you need to overstate your case to make a point. Obviously I don't find your explanation incomprehensible, but I do find it lacking. It is lacking at the very beginning, in that the entire point is how or why you think you know that you can always "add one". One other thing. Your statement at the end about "connecting such interests to rather elusive and incomprehensible doubts is pointless" is a subjective claim hidden as an oracular assertion about which there can be no dispute. You think it is pointless, no problem with that. You've been to Sunday School, and you've learned what you've been told. Good for you! Still, whether the doubts are indeed pointless or not is an entirely different matter. > Obviously I don't find your explanation incomprehensible, but I do > find it lacking. It is lacking at the very beginning, in that the > entire point is how or why you think you know that you can always "add > one". That's simply part of our conception of the naturals. I find the idea that some natural might -- perhaps by accident? -- lack a successor completely baffling, and can make nothing of it unless it is explained what such a thing might mean. > One other thing. Your statement at the end about "connecting such > interests to rather elusive and incomprehensible doubts is pointless" > is a subjective claim hidden as an oracular assertion about which > there can be no dispute. Anything at all can be disputed. > You think it is pointless, no problem with that. You've been to Sunday > School, and you've learned what you've been told. Good for you! Why do you think I've been to Sunday School? SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus On Nov 12, 7:36 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi> wrote: > > Obviously I don't find your explanation incomprehensible, but I do > > find it lacking. It is lacking at the very beginning, in that the > > entire point is how or why you think you know that you can always "add > > one". > > That's simply part of our conception of the naturals. This is just the ontological fallacy (from a property of a thing you can infer the existence of a thing). You say that you have a conception whereby a natural number always has a successor,. Fine; the only things that will be natural numbers are those things that have a successor. But that doesn't imply that there exist any natural numbers. Similarly, one could say that part our conception of God is that He is an absolutely perfect being. But that doesn't imply that there is any being who is absolutely perfect. I'd add that it seems to me worthwhile to distinguish our conception of what a natural number is, and our conception of what the natural- number sequence is. I think it is incorrect to say that part of our conception of what a natural number is is that it have a successor. We do have a conception of naturals, and we'd agree that 2 is a natural number; yet it just can't be, from the fact that 2 is a natural number, that 10^10^10^10 exists and is a natural number, which would in fact follow were every natural number always to have a successor (which is a natural). I'd agree with you that our conception of the natural-number sequence is that every natural in the sequence has a successor (in the sequence); but then the question just becomes whether there is any such sequence. > I find the idea that > some natural might -- perhaps by accident? -- lack a successor completely > baffling, and can make nothing of it unless it is explained what such a > thing might mean. Here's one way to picture it: after some very big point, the naturals fade away, ever so gradually. > > One other thing. Your statement at the end about "connecting such > > interests to rather elusive and incomprehensible doubts is pointless" > > is a subjective claim hidden as an oracular assertion about which > > there can be no dispute. > > Anything at all can be disputed. I agree with you. But surely you realize - because you surely you intend it - that your style tends at times to be oracular, where you assert something as if dispute is impossible. I usually find it more appropriate, for instance, to say, "I find it obvious that..." instead of "It's obvious that...". > > You think it is pointless, no problem with that. You've been to Sunday > > School, and you've learned what you've been told. Good for you! > > Why do you think I've been to Sunday School? Because you have? On Nov 12, 9:59 am, Aatu Koskensilta <aatu.koskensi...@xortec.fi> wrote: > > Somebody, who was in conversation with PS about this subject, asked > > him a question, "How do you know?" That would seem to be incentive [quoted text clipped - 27 lines] > from these or those (weak) principles -- but connecting such interests to > rather elusive and incomprehensible doubts is pointless. Right. So now the question is how do we reconcile the absolute certainty that PA is consistent with Goedel's theorem, which says that the consistency of PA is unprovable. It seems that you just proved it. You can prove it in ZFC? First of all I do not know if the ZFC proof is the same one as the manifest truth proof. Secondly, is ZFC consistent? > Right. So now the question is how do we reconcile the absolute > certainty that PA is consistent with Goedel's theorem, which says that > the consistency of PA is unprovable. It seems that you just proved it. Gödel's theorem does not imply that the consistency of PA is unprovable in any absolute sense, only that there is no formal derivation of "PA is consistent" in PA. > You can prove it in ZFC? First of all I do not know if the ZFC proof > is the same one as the manifest truth proof. In ZFC one would probably just show that the finite von Neumann ordinals are a model of PA. > Secondly, is ZFC consistent? Sure. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus On Nov 13, 1:21 am, Aatu Koskensilta <aatu.koskensi...@xortec.fi> wrote: > > Right. So now the question is how do we reconcile the absolute > > certainty that PA is consistent with Goedel's theorem, which says that [quoted text clipped - 3 lines] > any absolute sense, only that there is no formal derivation of "PA is > consistent" in PA. Thanks for the lecture but that does not help us to get around the problem. > > You can prove it in ZFC? First of all I do not know if the ZFC proof > > is the same one as the manifest truth proof. > > In ZFC one would probably just show that the finite von Neumann ordinals are > a model of PA. Is it the same as the manifest truth proof? > > Secondly, is ZFC consistent? > > Sure. How do we prove ZFC consistency? > -- > Aatu Koskensilta (aatu.koskensi...@xortec.fi) > > "Wovon man nicht sprechen kann, daruber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus >> Somebody, who was in conversation with PS about this subject, asked >> him a question, "How do you know?" That would seem to be incentive >> enough to provide at least a modicum of scrutiny or reflection. > > Surely you know the grounds on which we -- Peter, me, Torkel, and so on -- > find PA's consistency obvious by now. [...] > Now, if someone finds this explanation incomprehensible Who says they do? Perhaps they just find it 'too easy' as an answer? > I'm stumped. Glad that you admit that you don't understand the issue. > There's simply nothing I can do but shrug. Glad again that you admit it yourself. But the feeling might be more mutual than you think, you know. [Everything with a grain of salt, as usual.] SignatureCheers, Herman Jurjus > Who says they do? Perhaps they just find it 'too easy' as an answer? Given that the consistency of PA is an obvious triviality it should not be surprising the answer is easy. > Glad that you admit that you don't understand the issue. I do indeed find it utterly baffling people should worry about the consistency of PA, all the while accepting, apparently without any qualms, much more abstract mathematical statements unprovable in PA. There are exceptions, of course, such as Edward Nelson, whose objections and doubts are understandable and interesting, even if totally unrelated to the way the naturals are usually conceived, and the way we usually reason in mathematics. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus On Nov 13, 12:20 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi> wrote: There are > exceptions, of course, such as Edward Nelson, whose objections and doubts > are understandable and interesting, even if totally unrelated to the way the > naturals are usually conceived, and the way we usually reason in > mathematics. What are Nelson's objections and doubts which you understand? As near as I can tell, he complains about the Platonic existence of the naturals and then takes out his doubts on induction, all the while still assuming the existence of said naturals. > > > > On Nov 9, 4:17 am, aatu.koskensi...@xortec.fi wrote:> On 9 Oct, 10:20, Peter_Smith wrote: > [quoted text clipped - 23 lines] > Depends what M is: it if is a suitable set theory, by getting your > head around the idea of the structure of the iterative hierarchy. Let's assume then that M is a suitable set theory and that it is consistent. How do we prove that M is consistent? In a metatheory M_2? How do we know that M_2 is consistent? Which hierarchy did you have in mind, PA, M, M_2, M_3? If I get my head around this hierarchy M-omega does it mean that I am using a meta-meta-theory N? > (I know this might sound odd coming from someone whose day-job is as a > philosopher, but frankly, I do find "how we know?" questions are as > entirely boring applied to maths as applied to claims about medium- > sized dry goods. Scepticism either way is just uninteresting.)- I have not heard this one yet. I do not even quite understand what you are trying to say here. So let's just stick to the subjectmatter. "Newberry" <newberryxy@gmail.com> ha scritto >So let's confine ourselves to PA for now. We can prove that it is >consistent, that is we have proven G. How did we manage to do that >without running into a contradiction? We did not simply add Con(T) as >another axiom, we proved it. I suppose we proved it in some metatheory >M. How do we know that M is consistent? To know that theory (PA or M) is consistent we don't necessaily need a "proof". > "Newberry" <newberr...@gmail.com> ha scritto > [quoted text clipped - 6 lines] > To know that theory (PA or M) is consistent we don't necessaily need a > "proof". Right. So it means that Lucas and Penrose are rigtht, and Franzen is wrong? Newberry says... >> To know that theory (PA or M) is consistent we don't necessarily need a >> "proof". > >Right. So it means that Lucas and Penrose are right, and Franzen is >wrong? That doesn't follow at all. The question that Penrose asked was: Is it possible for there to be a computer program P(x) such that P(x) halts and returns true <-> x is the Godel number of a statement that human mathematicians can become convinced is an absolutely unassailable truth It is irrelevant whether the set of "unassailable truths" are provable or not. -- Daryl McCullough Ithaca, NY On Nov 13, 9:14 pm, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 15 lines] > It is irrelevant whether the set of "unassailable truths" > are provable or not. His point was that the human mind surpasses any machine. If the human mind can comprehend a truth that cannot be formally proven then the human mind surpasses any computer. Example: "if a set of axioms is manifestly true then the theory is consistent" is just as compelling as e.g. "~A, A v B |- B." It is not clear how any machine can prove that a theory based on manifestly true axioms is consistent. At least in this thread we failed to explain how it could be conclusively proven. "Newberry" <newberryxy@gmail.com> ha scritto > His point was that the human mind surpasses any machine. If the human > mind can comprehend a truth that cannot be formally proven then the > human mind surpasses any computer. Every truth can be formally proven. Just take the statement of the truth as an axiom. > Example: "if a set of axioms is manifestly true then the theory is > consistent" is just as compelling as e.g. "~A, A v B |- B." It is not > clear how any machine can prove that a theory based on manifestly true > axioms is consistent. At least in this thread we failed to explain how > it could be conclusively proven. I think ZFC can prove that any theory which has a model is consistent, and also can prove that any theory whose axioms are true in a model is consistent. Newberry says... >On Nov 13, 9:14 pm, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: [quoted text clipped - 21 lines] >mind can comprehend a truth that cannot be formally proven then the >human mind surpasses any computer. No, that doesn't follow at all. You're applying a double standard. You're only requiring that the human be able to "comprehend" a truth, while you're requiring that the computer be able to *prove* it. To make it a fair comparison, you either require both to prove the statement, or require neither to prove the statement. >Example: "if a set of axioms is manifestly true then the theory is >consistent" is just as compelling as e.g. "~A, A v B |- B." It is not >clear how any machine can prove that a theory based on manifestly true >axioms is consistent. Why does it matter whether the machine can prove it, if the human can't prove it, either? As I said, for a computer program to be as powerful as a human in recognizing truth, all that's necessary is for the program to be able to *recognize* true statements. It's not necessary that the program be able to *prove* them. Having said that, there is actually no problem in proving that a true theory must be consistent: Truth is preserved by logical deduction. A contradiction cannot be true. Therefore, it is impossible to deduce a contradiction from any true theory. -- Daryl McCullough Ithaca, NY On Nov 14, 4:56 pm, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 42 lines] > It's not necessary that the program be able to > *prove* them. How can a computer recognize that PA is consistent? > Having said that, there is actually no problem in proving > that a true theory must be consistent: Truth is preserved [quoted text clipped - 7 lines] > > - Show quoted text - Newberry says... >On Nov 14, 4:56 pm, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> Why does it matter whether the machine can prove it, >> if the human can't prove it, either? As I said, for [quoted text clipped - 5 lines] > >How can a computer recognize that PA is consistent? By producing an output "true" when given the input question "Do you believe that PA is consistent?". Are you asking how one would go about *programming* a computer program that would emulate a human's mathematical reasoning? If so, I have no idea. The issue is not whether we *currently* know how to make an artificial intelligent computer program. The issue is whether Godel's theorem implies that it is impossible. It doesn't imply any such thing. There is what I think is a pretty air-tight argument that no single human can do any mathematical reasoning that is noncomputable: A real human has a finite memory capacity, and so there are only finitely many different statements of mathematics that we can ever hold in our heads at one time. So the collection of all statements that any *actual* human would ever claim to be "unassailably true" is a finite set. Every finite set of formulas is computable. Now, you could argue about what an idealized human could do, where we idealize the human to have an infinite memory capacity. Could such an idealized human do something that no Turing machine could do? Well, it depends on the details of how the "ideal human" is idealized. -- Daryl McCullough Ithaca, NY On Nov 14, 9:13 pm, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 32 lines] > is a finite set. Every finite set of formulas > is computable. Bingo!! You got it! So we have the human mind surpasses any machine and no single human can do any mathematical reasoning that is noncomputable. A contradiction! That is what I was trying to say all along. > Now, you could argue about what an idealized > human could do, where we idealize the human [quoted text clipped - 7 lines] > Daryl McCullough > Ithaca, NY Newberry says... >On Nov 14, 9:13 pm, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> There is what I think is a pretty air-tight argument >> that no single human can do any mathematical reasoning [quoted text clipped - 11 lines] >noncomputable. A contradiction! That is what I was trying to say all >along. No, we *don't* have that the human mind surpasses any machine. There is no reason to believe that's true. -- Daryl McCullough Ithaca, NY On Nov 15, 3:12 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 18 lines] > No, we *don't* have that the human mind surpasses any machine. > There is no reason to believe that's true. But we do. We know that G is true. Proof: The axioms of PA are manifestly true PA is consistent "PA is consistent" is equivalent to G G QED This proof cannot be formalized. We can prove the consistency of PA in ZFC. We believe ZFC to be true. We can prove the consistency of ZFC only in a stronger theory about which we are not sure that it is true. We observe two things 1) This proof is not the same as the manifest truth proof 2) Since we have not proved the consistenvy of ZFC we do not know if the proof of the consistency of PA is not a falsehood. So our certainty that PA is true does not come from this ZFC based proof. If the manifest truth proof were formalizable and the axioms/rules were added to PA we would have a contradiction. Hence it is not formalizable i.e. the human mind surpasses any machine. Newberry says... >On Nov 15, 3:12 am, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> No, we *don't* have that the human mind surpasses any machine. >> There is no reason to believe that's true. > >But we do. We know that G is true. That does not prove that the human mind surpasses any machine. You can certainly program a machine to know, among its basic facts, that G(PA) is true. >Proof: >The axioms of PA are manifestly true [quoted text clipped - 3 lines] > >This proof cannot be formalized. That's just not true. It can perfectly well be formalized. It just can't be formalized in the language of PA. Extend PA to a new theory PA-plus in the following way: Add a new predicate symbol T(x). For every statement S in the language of PA, add the axiom S <-> T(#S) where #S means the Godel code of S. Then add the axiom Ax Prove(PA,x) -> T(x) where Prove(PA,x) is the formalization of the proof predicate. In PA-plus, it is perfectly straight-forward to prove G(PA) (the Godel statement for PA). -- Daryl McCullough Ithaca, NY On Nov 15, 9:10 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 38 lines] > In PA-plus, it is perfectly straight-forward to prove > G(PA) (the Godel statement for PA). A couple of comments. Firstly, I do not know if the proof you are showing is the same as the manifest truth proof. Secondly, if I understand you correctly the only difference between (Ex)P(x, #("F") --> F and Ax Prove(PA,x) -> T(x) is the "T." It is not that it cannot be formalized in the language of PA, it is that it leads to a contradiction. Did we put the axiom on a meta-level just to avoid the contradiction? BTW, I do not understand this: S <-> T(#S), where #S means the Godel code of S. The Goedel number of S is true ... ? Newberry says... >A couple of comments. Firstly, I do not know if the proof you are >showing is the same as the manifest truth proof. I have no idea what you mean by "the manifest truth proof", unless it is just the informal argument: Every axiom of PA is true. Truth is preserved under logical deduction. Therefore, every theorem of PA is true. A contradiction can't be true. Therefore, PA can't prove any contradictions. This proof can certainly be formalized. It just can't be formalized in PA. >Secondly, if I >understand you correctly the only difference between (Ex)P(x, #("F") >--> F and Ax Prove(PA,x) -> T(x) is the "T." It is not that it cannot >be formalized in the language of PA, it is that it leads to a >contradiction. No, it can't be formalized in PA, because there is no formula in PA which says "S is a true sentence of arithmetic". >Did we put the axiom on a meta-level just to avoid the >contradiction? We put it in at the meta-level because PA doesn't have a truth predicate. >BTW, I do not understand this: >S <-> T(#S), where #S means the Godel code of S. >The Goedel number of S is true ... ? No, T(#S) doesn't say #S is true, it says "#S is the Godel code of a true statement". -- Daryl McCullough Ithaca, NY On Nov 16, 8:07 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 12 lines] > This proof can certainly be formalized. It just > can't be formalized in PA. 1) We can see that every axiom of PA is true. In you proof it is probably just an assumption. 2) The proof can be formalized in ZFC but we do not know if ZFC is consistent. So it may prove falsehoods. > >Secondly, if I > >understand you correctly the only difference between (Ex)P(x, #("F") [quoted text clipped - 4 lines] > No, it can't be formalized in PA, because there is no formula > in PA which says "S is a true sentence of arithmetic". What I meant is that we can say (Ex)(Px, #(F)) --> F leaving the T out. Yes, it is inconsistent. That is my point. But you claimed it was not the reason we put (Ex)(Px, #(F)) --> T(F) on the meta-level. > >Did we put the axiom on a meta-level just to avoid the > >contradiction? [quoted text clipped - 8 lines] > No, T(#S) doesn't say #S is true, it says "#S is the > Godel code of a true statement". What does it say then? > -- > Daryl McCullough > Ithaca, NY Newberry says... >On Nov 16, 8:07 am, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> I have no idea what you mean by "the manifest truth proof", >> unless it is just the informal argument: [quoted text clipped - 13 lines] >2) The proof can be formalized in ZFC but we do not know if ZFC is >consistent. So it may prove falsehoods. So? >What I meant is that we can say >(Ex)(Px, #(F)) --> F >leaving the T out. >Yes, it is inconsistent. No, it's not inconsistent, if you are careful about it. >> >Did we put the axiom on a meta-level just to avoid the >> >contradiction? [quoted text clipped - 10 lines] > >What does it say then? I just said it: T(#S) says "S is the code of a true statement". -- Daryl McCullough Ithaca, NY On Nov 17, 10:26 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 26 lines] > > No, it's not inconsistent, if you are careful about it. Please explain. You were the one who claimed it was inconsistent in an earlier post. > >> >Did we put the axiom on a meta-level just to avoid the > >> >contradiction? [quoted text clipped - 18 lines] > > - Show quoted text - Newberry says... >On Nov 17, 10:26 am, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> >What I meant is that we can say >> >(Ex)(Px, #(F)) --> F [quoted text clipped - 5 lines] >Please explain. You were the one who claimed it was inconsistent in an >earlier post. We talked about a bunch of different things. I don't know what P is supposed to mean here. Is it the provability predicate? If so, the provability predicate for what theory? What is F? Is it just any contradiction? I'll assume it is. If you have a theory T1, then you can define a provability predicate for T1, call it P1(x,y) (meaning "x is a code for a proof in T1 of a formula whose code is y"). If T1 is a sound theory (anything it says about arithmetic is true in the usual interpretation), then the statement Ex (P1(x,#F)) -> F is a perfectly true sentence. So you can add that as an axiom to T1 to get a new theory T2. There is no problem with consistency. T2 can prove G1, the Godel sentence for T1. But T2 cannot prove G2, the Godel sentence for T2. You can define a provability predicate for T2, call it P2, and you can formulate a perfectly good statement of arithmetic: Ex (P2(x,#F)) -> F You can add this as an axiom to T2 to get a new theory T3. And so on. This process gives you a way to go from axiomatizable true theories to more complete axiomatizable true theories. Now, if you want to go for the whole ball of wax and come up with a theory T_ultimate with the following property: T_ultimate proves Ex (P_ultimate(x,#F)) -> F There is no such theory T_ultimate except for an inconsistent theory. -- Daryl McCullough Ithaca, NY On Nov 17, 5:01 pm, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 48 lines] > There is no such theory T_ultimate except for an > inconsistent theory. And this is not true. On Nov 17, 5:01 pm, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 48 lines] > There is no such theory T_ultimate except for an > inconsistent theory. Do you mean that there is no such extension of PA (classical logic with Peano axioms) or do you mean there is no such extension of ANY theory capable of arithmetic. If you mean the later then I think it is not true. Newberry says... >On Nov 17, 5:01 pm, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> Now, if you want to go for the whole ball of wax and >> come up with a theory T_ultimate with the following [quoted text clipped - 9 lines] >with Peano axioms) or do you mean there is no such extension of ANY >theory capable of arithmetic. I mean the latter, but I'm not sure about the distinction you are making. It's hard to see how something could be "capable of arithmetic" without being an extension of PA (or at least an extension of Robinson arithmetic). >If you mean the later then I think it is >not true. Yes, it is. That's what Godel's second incompleteness theorem proves: If T_ultimate is consistent, then it cannot prove its own consistency. -- Daryl McCullough Ithaca, NY On Nov 19, 11:34 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 18 lines] > of arithmetic" without being an extension of PA (or at least > an extension of Robinson arithmetic). When you say "arithmetic" what do you mean by that? And what do you mean by "PA"? "Peano arithmetic" or a specific formal system = classical logic + Peano axioms"? What if we do not use classical logic? Is it still Peano arithmetic? Is it still "arithmetic"? > >If you mean the later then I think it is > >not true. > > Yes, it is. That's what Godel's second incompleteness theorem > proves: If T_ultimate is consistent, then it cannot prove its > own consistency. Even Aatu Koskensilta posted a contribution not to long ago where he concedes that the 2-nd theorem depends on the particulars of the formal system. I have never seen a proof of "Any conceivable formal system capable of arithmetic (representing all p.r. functions) cannot prove its own consistency." Newberry says... >On Nov 19, 11:34 am, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> >Do you mean that there is no such extension of PA (classical logic >> >with Peano axioms) or do you mean there is no such extension of ANY [quoted text clipped - 6 lines] > >When you say "arithmetic" what do you mean by that? For me, to say that a theory is capable of arithmetic means that it can prove the following theorems: 1. Ax x+0 = x 2. Ax Ay x+(y+1) = (x+y) + 1 3. Ax x * 0 = 0 4. Ax x * (y+1) = (x*y) + x 5. Ax ~ ((x+1)=0) 6. Ax Ay ((x+1) = (y+1) -> x=y) >And what do you mean by "PA"? Basically, axioms 1-6 together with the induction schema: Phi(0) & (Ax (Phi(x) -> Phi(x+1))) -> Ax (Phi(x)) >"Peano arithmetic" or a specific formal system = >classical logic + Peano axioms"? What if we do not use classical >logic? Is it still Peano arithmetic? Is it still "arithmetic"? You can use the same axioms for intuitionistic arithmetic. The distinction is not really important for discussing Godel's theorem. >> >If you mean the later then I think it is >> >not true. [quoted text clipped - 6 lines] >concedes that the 2-nd theorem depends on the particulars of the >formal system. >I have never seen a proof of "Any conceivable formal >system capable of arithmetic (representing all p.r. functions) >cannot prove its own consistency." I'm not sure what Aatu meant, but if you define consistency of a theory T via the provability predicate, Con(#T) == Ax ~Pr(#T,x,#0=1) "there does not exist a proof of 0=1 from the axioms of T", then if T is capable of proving some routine facts about arithmetic (I'm not sure whether induction is needed or not) then it follows that if T proves Con(#T), then T is inconsistent. -- Daryl McCullough Ithaca, NY > Newberry says... > [quoted text clipped - 26 lines] > > Phi(0) & (Ax (Phi(x) -> Phi(x+1))) -> Ax (Phi(x)) Do we have to prove 1+0 = 1 v 1 = 0 ? (1) or ~(1+0 # 1 & 1 = 1) (2) ? What if we are able to prove only 1+0 = 1 (3) ~(1+0 # 1) (4) but not (1), (2)? Is it still Peano arithmetic? > >"Peano arithmetic" or a specific formal system = > >classical logic + Peano axioms"? What if we do not use classical [quoted text clipped - 31 lines] > Daryl McCullough > Ithaca, NY Newberry says... >> Newberry says... >> [quoted text clipped - 32 lines] >~(1+0 # 1 & 1 = 1) (2) >? You don't have to prove anything. I don't understand what you are asking. >What if we are able to prove only >1+0 = 1 (3) >~(1+0 # 1) (4) >but not (1), (2)? Is it still Peano arithmetic? As I said, Peano arithmetic is a specific theory. I suppose there are minor variants that would still be considered Peano arithmetic, but giving up (1) seems pretty major to me. But I don't understand what you are getting at. -- Daryl McCullough Ithaca, NY On Nov 21, 6:27 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 49 lines] > > But I don't understand what you are getting at. The question is if any conveivable theory capable of artithmetic is unable to prove its own consistency. Let's say we give up (1) and (2) and they become undecidable. OK, let's not call it Peano arithmetic. But why exactly do we need (1), (2)? And let's take it a step further. If for some predicate F we can prove ~(Ex)Fx do we also have to prove ~(Ex)(Fx & Gx)? For example if we prove ~(Ex)(x+0 # x) (5) do we also need to prove ~(Ex)((x+0 # x) & (x = x)) (6) ? When we are able to prove (5) why do we have to prove (6)? Will we lose any knowledhe of arithmetic if (6) is undecidable? > Even Aatu Koskensilta posted a contribution not to long ago where he > concedes that the 2-nd theorem depends on the particulars of the > formal system. Of course it does: the second incompleteness theorem applies to theories satisfying the Hilbert-Bernays-Loeb derivability conditions. This has nothing to do with your confused ramblings on Lucas's and Penrose's arguments. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus On Nov 17, 5:01 pm, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 52 lines] > Daryl McCullough > Ithaca, NY This is interesting stuff. Where can I read about it? How do you construct P_ultimate(x,y)? Newberry says... >> Now, if you want to go for the whole ball of wax and >> come up with a theory T_ultimate with the following [quoted text clipped - 5 lines] >> There is no such theory T_ultimate except for an >> inconsistent theory. >This is interesting stuff. Where can I read about it? How do you >construct P_ultimate(x,y)? Well, as I said, it's inconsistent, so it's not really that interesting. But you can do this: Let Pr(x,y,z) be defined so that it holds if and only if x is a code for an r.e. theory T, and y is a code for a proof p, and z is a code for a formula S, and p is a proof of S from the axioms of T. If #T0 is the code for an r.e. set of axioms in the language of PA, then let f(#T0) be the code for the theory T1 where the axioms of T1 consists of the axioms of T0 plus the "soundness" schema for T0, which is, for every formula S in the language of PA, Ex Pr(#T0,x,#S) -> S Now we just use the fixed point theorem for r.e. sets to come up with a theory T extending PA such that #T and f(#T) code the same r.e. set of axioms. Here's what the fixed point theorem for r.e. sets says: If x is any natural number, then let W_x be the r.e. set coded by x (or the empty set, if x doesn't code anything). Then if f is any recursive function from naturals to naturals, then there is a natural number n such that W_n = W_{f(n)} -- Daryl McCullough Ithaca, NY On Nov 17, 5:01 pm, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 48 lines] > There is no such theory T_ultimate except for an > inconsistent theory. Do you agree though that adding Ex (P_ultimate(x,#F)) -> F would force the theory in the standard model? On Nov 15, 6:13 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > There is what I think is a pretty air-tight argument > that no single human can do any mathematical reasoning [quoted text clipped - 6 lines] > is a finite set. Every finite set of formulas > is computable. A finite machine can't compute the palindrome or multiplication function, so it would seem by the same token neither any actual human nor any actual computer can "do" multiplication. This strikes me as perhaps not what people have in mind. abo says... >A finite machine can't compute the palindrome or multiplication >function, so it would seem by the same token neither any actual human >nor any actual computer can "do" multiplication. I think that's all perfectly true. We can't multiple two trillion-digit numbers. We can't compute the palindrome of a trillion-letter word. -- Daryl McCullough Ithaca, NY Daryl McCullough says... >abo says... > [quoted text clipped - 5 lines] >trillion-digit numbers. We can't compute the palindrome >of a trillion-letter word. Whoops! What I meant was that we can't decide whether a trillion-letter word is a palindrome. -- Daryl McCullough Ithaca, NY > A finite machine can't compute the palindrome or multiplication > function, so it would seem by the same token neither any actual human > nor any actual computer can "do" multiplication. Right. When saying that humans and computers can do multiplication we have in mind a perfectly clear idealised picture, and can explain that we mean by this ability that we've an explicit algorithm for carrying our the task, and obviously only limitation of time and space prevent actual humans and computers carrying out multiplications with arbitrary large figures. So what idealised picture could be involved in talking about truths that a human would ever claim to be "unassailably true"? Unless that question is answered it is utterly obscure what counts as an "unassailable truth". > This strikes me as perhaps not what people have in mind. Probably not. It's up to them to explain what it is they have in mind if anything is to be made of claims about what humans would -- "in principle" -- find acceptable, unassailable, and so on. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus Aatu Koskensilta says... >Right. When saying that humans and computers can do multiplication we have >in mind a perfectly clear idealised picture, and can explain that we mean by [quoted text clipped - 4 lines] >would ever claim to be "unassailably true"? Unless that question is answered >it is utterly obscure what counts as an "unassailable truth". Well, to be fair, while Roger Penrose didn't define what "unassailably true" means, he did characterize it. Basically, it includes the following: 1. Any proof using standard mathematics (say, ZFC) is unassailably true. 2. If it is unassailably true that all the axioms of theory T are unassailably true statements, then any theorem of T is unassailably true. 3. No contradiction is unassailably true. 4. If any statement is unassailably true, then the fact that it is unassailably true is unassailably true. -- Daryl McCullough Ithaca, NY On Nov 19, 8:08 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi> wrote: > > A finite machine can't compute the palindrome or multiplication > > function, so it would seem by the same token neither any actual human [quoted text clipped - 7 lines] > idealised picture could be involved in talking about truths that a human > would ever claim to be "unassailably true"? I'm not sure there is an idealised picture, or whether there is any connection between an idealized picture and unassailable truth. I have no money on this particular horse in this thread. What I was replying to was Daryl's argument that "no single human can do any mathematical reasoning that is noncomputable." The fact (if not a fact, anyway I think it's true) that single humans are limited to a finite number of beliefs does not really have a bearing on whether their mathematical reasoning is non-computable. Sure in some sense it's computable because any of their actual reasonings are finite; but that's not I think really what's under discussion when someone suggests that the reasoning is non-computable. > it is utterly obscure what counts as an "unassailable truth". > [quoted text clipped - 3 lines] > anything is to be made of claims about what humans would -- "in principle" > -- find acceptable, unassailable, and so on. Again, I have no money on the unassailable horse in this thread. I wanted to reply to the claim that mathematical reasoning by any particular human is computable; I don't think this is really follows from Daryl's argument. > -- > Aatu Koskensilta (aatu.koskensi...@xortec.fi) > > "Wovon man nicht sprechen kann, daruber muss man schweigen" > - Ludwig Wittgenstein, Tractatus Logico-Philosophicus On Nov 14, 9:13 pm, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 16 lines] > a computer program that would emulate a human's > mathematical reasoning? If so, I have no idea. Easy. Just add this axiom: (Ex)P(x, #("F") --> F > The issue is not whether we *currently* know > how to make an artificial intelligent computer [quoted text clipped - 24 lines] > Daryl McCullough > Ithaca, NY Newberry says... >On Nov 14, 9:13 pm, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> Are you asking how one would go about *programming* >> a computer program that would emulate a human's >> mathematical reasoning? If so, I have no idea. > >Easy. Just add this axiom: (Ex)P(x, #("F") --> F But Godel shows that adding that axiom is inconsistent. -- Daryl McCullough Ithaca, NY On Nov 15, 8:52 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 7 lines] > > But Godel shows that adding that axiom is inconsistent. I do not know if Goedel actually showed this particular case but it certinly is inconsistent. But what is wrong with this axiom? It is just as compelling as the rest of them. When people say that G is true because of its meaning this is the derivation rule they use. > On Nov 15, 8:52 am, stevendaryl3...@yahoo.com (Daryl McCullough) > wrote: [quoted text clipped - 15 lines] > just as compelling as the rest of them. When people say that G is true > because of its meaning this is the derivation rule they use. What people? Where does anyone use such a rule? MoeBle Newberry says... >On Nov 15, 8:52 am, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> Newberry says... >> >Easy. Just add this axiom: (Ex)P(x, #("F") --> F >> [quoted text clipped - 4 lines] >just as compelling as the rest of them. When people say that G is true >because of its meaning this is the derivation rule they use. It depends on what P is. If you create a program P that proves theorems, and you have good reason to believe that P is consistent, then you can create a new theory T with the axiom schema (P proves S) -> S Program P itself can't take advantage of this axiom without being inconsistent, but T can. T will necessary go beyond what P can prove. -- Daryl McCullough Ithaca, NY On Nov 15, 7:33 pm, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 19 lines] > being inconsistent, but T can. T will necessary go beyond > what P can prove. Here we go again - a meta-level. This looks a little bit like Russell's theory of types. But the theory of types has its own difficulties. Does T surpass the machine P? BTW, I do not see any reason why the axiom schema should be on another level. It is just as compelling as the rest of them. We do not split Peano's axioms either and do not say these axioms hold in P, you can add two more in S but P cannot take any advantage of them. There is a difference between saying a theory or machine is intrinsically incapable of proving some true sentences and saying the machine "cannot" prove them because a contradiction would result. So we patch the system by creating an ad hoc meta-level. "Newberry" <newberryxy@gmail.com> ha scritto >> It depends on what P is. If you create a program P that proves >> theorems, and you have good reason to believe that P is consistent, [quoted text clipped - 14 lines] > Peano's axioms either and do not say these axioms hold in P, you can > add two more in S but P cannot take any advantage of them. There is no other levels. The statement G="(P proves S) -> S" is an arithmetical statement. You can of course add it to the axioms of P, but once you have done it you get another set of axioms, not P anymore. You can also consider the statement G'="(P+*this statement* proves S)->S" and add it to P. So you obtain a theory P' including the axiom "(P' proves S)->S". But the statement G' is false and can be proven to be false in P. So if you add it to P you obtain an inconsistent theory. Newberry says... >On Nov 15, 7:33 pm, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> If you create a program P that proves >> theorems, and you have good reason to believe that P is consistent, [quoted text clipped - 8 lines] >Here we go again - a meta-level. This looks a little bit like >Russell's theory of types. No, not really. The types are not different; it can all be done in the language of arithmetic, where the only type is "natural number". It's more like Tarski's levels of truth. There is no problem having a truth predicate for arithmetic in a language that goes beyond arithmetic. But you can't have a truth predicate for arithmetic *within* arithmetic. >But the theory of types has its own >difficulties. Does T surpass the machine P? That's what I just said: "T will necessarily go beyond what P can prove". >BTW, I do not see any reason why the axiom schema should be on another >level. What do you mean by "on another level"? Let me just introduce a bit of terminology. The axiom schema "(P proves S) -> S" is the "soundness schema" for P. It says that P is sound, in the sense that it never proves a false statement. So the way I've defined T, T proves the soundness schema for P. But T *cannot* prove its own soundness schema. There is no consistent theory T (at least not with ordinary first-order logic) that has an axiom saying (T proves S) -> S >It is just as compelling as the rest of them. What is just as compelling? >We do not split Peano's axioms either and do not say these >axioms hold in P, you can add two more in S but P cannot >take any advantage of them. PA is *already* a specific, well-defined theory. If you add axioms to it, you have formed a *new* theory, PA-plus. Of course PA can't take advantage of these new axioms, because they aren't axioms of PA. The same is true of a program P that proves theorems. If P is sound, then the soundness schema for P is true, but is not provable by P. You can certainly *modify* P by adding the soundness schema, but then you've created a *new* program, P'. >There is a difference between saying a theory or machine is >intrinsically incapable of proving some true sentences and saying the >machine "cannot" prove them because a contradiction would result. It's not that the machine cannot prove them *because* a contradiction would result. Machines don't care whether they prove a contradiction or not. Rather, the implication is this: *if* you construct a theory that can prove its own soundness schema, *then* that theory is inconsistent. -- Daryl McCullough Ithaca, NY On Nov 16, 7:57 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 18 lines] > > It's more like Tarski's levels of truth. OK, Tarski's truth level. Whatever. Same thing. There is no problem > having a truth predicate for arithmetic in a language that > goes beyond arithmetic. But you can't have a truth predicate > for arithmetic *within* arithmetic. If you say "can't" what does the impossibility consist of. Is it somehow inherently impossible or it "can't" be done because a contradiction would result? > >But the theory of types has its own > >difficulties. Does T surpass the machine P? [quoted text clipped - 23 lines] > > What is just as compelling? This: T(F) --> F > >We do not split Peano's axioms either and do not say these > >axioms hold in P, you can add two more in S but P cannot [quoted text clipped - 26 lines] > Daryl McCullough > Ithaca, NY Newberry says... >On Nov 16, 7:57 am, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: > There is no problem >> having a truth predicate for arithmetic in a language that [quoted text clipped - 4 lines] >somehow inherently impossible or it "can't" be done because a >contradiction would result? I already answered that question: >> There >> is no consistent theory T (at least not with ordinary >> first-order logic) that has an axiom saying >> >> (T proves S) -> S -- Daryl McCullough Ithaca, NY On Nov 17, 10:32 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 20 lines] > Daryl McCullough > Ithaca, NY OK, so now we are getting somewhere. By using eiher (Ex)(Px, #(F)) --> F or (Ex)(Px, #(F)) --> T(F) we can perfectly prove either. But since the former leads to a contradiction we opted for the second - Tarski's truth levels. But there is one problem: T(F) --> F is just as compelling as any other axiom. But there is one problem. This is just as compelling as any other axiom: T(F) --> F Newberry says... >On Nov 17, 10:32 am, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> >> There >> >> is no consistent theory T (at least not with ordinary >> >> first-order logic) that has an axiom saying >> >> >> (T proves S) -> S >OK, so now we are getting somewhere. By using eiher >(Ex)(Px, #(F)) --> F I'm sorry, I don't remember what definition you are using for "P". I assume it's a proof predicate: P(x,y) means that x is a proof of the statement whose code is y. But what *theory* is used to prove y? And what is F? Is it a false statement, or what? >or >(Ex)(Px, #(F)) --> T(F) >we can perfectly prove either. What is T(F)? >But since the former leads to a >contradiction we opted for the second - Tarski's truth levels. >But there is one problem: >T(F) --> F >is just as compelling as any other axiom. The problem here is that you haven't defined what the heck you are talking about. What is T(F)? How about defining your terms before using them? T(F) --> F isn't compelling to me, because I don't even know what it means. -- Daryl McCullough Ithaca, NY On Nov 17, 4:47 pm, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 39 lines] > Daryl McCullough > Ithaca, NY P(x,y) is a provability predicate F is any wff T(F) is true T(F) --> F is an axiom schema Newberry says... >On Nov 17, 4:47 pm, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> The problem here is that you haven't defined what >> the heck you are talking about. What is T(F)? [quoted text clipped - 4 lines] >> isn't compelling to me, because I don't even >> know what it means. >P(x,y) is a provability predicate For what theory? >F is any wff >T(F) is true How are you defining T(F)? Tarski showed that no consistent language can have a truth predicate for that very language. -- Daryl McCullough Ithaca, NY On Nov 19, 11:38 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 11 lines] > > For what theory? Say PA. > >F is any wff > >T(F) is true > > How are you defining T(F)? Tarski showed that no consistent > language can have a truth predicate for that very language. I am not defining it. You defined it in your previous posts. I did not say that it was in the language of PA. I said that T(F) --> F is just as compelling as any other axiom. What I mean is that by puting T(F) on another level you did avoid a contradiction, nevertheless intuitively T(F) --> F. ("F" means any wff not "false."} Newberry says... >On Nov 19, 11:38 am, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> How are you defining T(F)? Tarski showed that no consistent >> language can have a truth predicate for that very language. > >I am not defining it. You defined it in your previous posts. No, I didn't. I used the symbol T to mean a particular theory. So by T(F) do you mean "T proves F"? >I did not say that it was in the language of PA. I said that >T(F) --> F >is just as compelling as any other axiom. Axiom for *what*? Yes, if you start with a theory T, you can use the soundness schema for T as an axiom for a new theory T1. You can't use it as an axiom schema for T, because that's ill-defined. You have to know what T is before you can add T(F) -> F as an axiom. >What I mean is that by puting T(F) on another level >you did avoid a contradiction, nevertheless intuitively >T(F) --> F. ("F" means any wff not "false."} Yes, if T is sound, then T(F) -> F (that's what "sound" means). There is no problem with adding such an axiom. -- Daryl McCullough Ithaca, NY > Newberry says... > [quoted text clipped - 7 lines] > No, I didn't. I used the symbol T to mean a particular theory. > So by T(F) do you mean "T proves F"? "T(F)" means "F is true" > >I did not say that it was in the language of PA. I said that > >T(F) --> F > >is just as compelling as any other axiom. > > Axiom for *what*? If a sentence F is true then F. Yes, if you start with a theory T, > you can use the soundness schema for T as an axiom for > a new theory T1. You can't use it as an axiom schema for [quoted text clipped - 11 lines] > Daryl McCullough > Ithaca, NY Newberry says... >"T(F)" means "F is true" Well, there is no such formula that means "F is true" for *arbitrary* F. You can restrict yourself to a specific language L and introduce a predicate T(F) with the interpration that T(F) holds if F is a true formula of language L, but as proved by Tarski, the predicate T cannot be in the language L. >> >I did not say that it was in the language of PA. I said that >> >T(F) --> F [quoted text clipped - 3 lines] > >If a sentence F is true then F. I meant: axiom for what theory? -- Daryl McCullough Ithaca, NY "Daryl McCullough" <stevendaryl3016@yahoo.com> ha scritto >>"T(F)" means "F is true" > [quoted text clipped - 4 lines] > as proved by Tarski, the predicate T cannot be in the language > L. Are you saying that - adding to the language L of PA a unary relation symbol "T" to obtain the language L' - adding to PA the axiom schemata "T(#F)->F" for any wff F of L' we get an inconsistent theory? LordBeotian says... >"Daryl McCullough" <stevendaryl3016@yahoo.com> ha scritto > [quoted text clipped - 12 lines] >- adding to PA the axiom schemata "T(#F)->F" for any wff F of L' >we get an inconsistent theory? Actually, that schema by itself is consistent. That schema doesn't say that T is the truth predicate. It only says that T is sound. You need T(#F) <-> F in order for T to be the truth predicate. Adding that schema leads to an inconsistency. -- Daryl McCullough Ithaca, NY > Newberry says... > [quoted text clipped - 6 lines] > as proved by Tarski, the predicate T cannot be in the language > L. I never said it was in the language of L. It is in the meta-langage. > >> >I did not say that it was in the language of PA. I said that > >> >T(F) --> F [quoted text clipped - 5 lines] > > I meant: axiom for what theory? For any sentence in any theory "if a sentence F is true then F" is intuitively self-evident. It is at least as compelling as "truth cannot be inconsistent with itself." Newberry says... >> Well, there is no such formula that means "F is true" >> for *arbitrary* F. You can restrict yourself to a specific [quoted text clipped - 4 lines] > >I never said it was in the language of L. It is in the meta-langage. Well, in the meta-language the schema is just plain vaccuously true. >For any sentence in any theory "if a sentence F is true then F" is >intuitively self-evident. I would say that it is true by definition. "F is true" means the same thing as F. But now I've lost track of what you are claiming follows from this schema. -- Daryl McCullough Ithaca, NY It would take too long to separate out the various confusions in this thread from the occasional bits of informed good sense. But it might be worth pointing out to anyone who is actually *seriously* interested in these issues -- rather than in just sounding off on the basis of half-understood ideas -- that there is actually an extensive literature on exactly what happens when you add truth-theories of different strengths to arithmetic. Start by googling for work by e.g. Volker Halbach or Jeffrey Ketland (for fairly accessible treatments). Worth doing, as some of results -- e.g. about the proof-theoretic strength of what can look to be modest augmentations of first-order PA -- are moderately surprising. > It would take too long to separate out the various confusions in this > thread from the occasional bits of informed good sense. But it might [quoted text clipped - 7 lines] > strength of what can look to be modest augmentations of first-order PA > -- are moderately surprising. Do they deal with this? QUOTE: Ex (P1(x,#F)) -> F Ex (P2(x,#F)) -> F Ex (P_ultimate(x,#F)) -> F There is no such theory T_ultimate except for an inconsistent theory. END OF QUOTE One thing is apparent about these theories: They are either inconclusive or inconsistent. > > It would take too long to separate out the various confusions in this > > thread from the occasional bits of informed good sense. But it might [quoted text clipped - 9 lines] > > Do they deal with this? It isn't my job to make up for your not keeping up with the literature and trying to make it up as you go along. Read the stuff and do the work. On Nov 21, 6:23 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 16 lines] > same thing as F. But now I've lost track of what you are claiming > follows from this schema. I am claiming that by proving that G is true we have proven G. If we have proven this in a metalanguage then this system has its own G_1 and its consistency is unprovable other than in even higher mata- language. Therefore we have no consistency proof of PA at all. Then there is the manifest truth proof: The axioms of PA are manifestly true PA derivations preserve truth Truth cannot be inconsistent with itself i.e. PA is consistent This proof is absolute and conclusive. It means that it is not formalizable. For if it were it would either not be conclusive just like in the meta-theory above or we would have an inconsistency just as if we used P_universal(x,y). It means that the human mind surpasses any machine. How did we manage to surpass any machine? Since the proof is conclusive we must have accomplished some non-formalizable analogy of P_universal. Since it is not formalizable we are off the hook. We can eat the turkey and have it. We have proven G without running into a contradiction. You know what I am getting at? Newberry says... >I am claiming that by proving that G is true we have proven G. Okay. But recall what G is. For any theory T capable of coding up sufficiently much of proof theory, you can form a sentence G such that G <-> not Pr_T(#G) where Pr_T(x) is the formalization in T of "x is the code of a statement provable in T". In other words, G is true if and only if G is not provable by theory T. It may be provable by *other* theories. >If we have proven this in a metalanguage then this system has its own G_1 >and its consistency is unprovable other than in even higher mata- >language. Therefore we have no consistency proof of PA at all. A proof is relative to a set of axioms. There are axioms that are strong enough to prove the consistency of PA, so your claim is false. If you are claiming that it doesn't count unless the theory can prove its own axioms to be consistent, well Godel showed that there is no system that can do that. >Then there is the manifest truth proof: >The axioms of PA are manifestly true >PA derivations preserve truth >Truth cannot be inconsistent with itself >i.e. PA is consistent >This proof is absolute and conclusive. It's an appeal to our intuitions. We don't prove the correctness of those intuitions, any more than ZFC proves the consistency of the axioms of ZFC. >It means that it is not formalizable. Well, we've gone into a complete circle. I say it's perfectly well formalizable. >For if it were it would either >not be conclusive It's *not* conclusive in the sense that you are wanting things to be conclusive. Nothing is. >just like in the meta-theory above or we would have >an inconsistency just as if we used P_universal(x,y). It means that >the human mind surpasses any machine. No, it doesn't. >How did we manage to surpass any machine? Since the proof is >conclusive we must have accomplished some non-formalizable analogy of [quoted text clipped - 3 lines] > >You know what I am getting at? Yes, but it is completely wrong. There is no sense in which we can prove the Godel sentence for PA in any more conclusive fashion than a machine can. -- Daryl McCullough Ithaca, NY terrific good sense :-))))))))))))))))) On Nov 24, 7:12 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 17 lines] > > A proof is relative to a set of axioms. "PA is consistent" means that in PA, for any P we will never derive P & ~P no matter how long and in what order we keep generating theorems. It is not relative to any axioms. It either is the case or not. This is what we are interested in proving. There are axioms that > are strong enough to prove the consistency of PA, so your claim > is false. If you are claiming that it doesn't count unless the > theory can prove its own axioms to be consistent, well Godel > showed that there is no system that can do that. So how can we possible arrive at the conclusion that PA is consistent? > >Then there is the manifest truth proof: > >The axioms of PA are manifestly true [quoted text clipped - 12 lines] > Well, we've gone into a complete circle. I say > it's perfectly well formalizable. So which one is it? A) We don't prove the correctness of those intuitions B) it's perfectly well formalizable We can even try one hypothesis after another but not both at the same time. > >For if it were it would either > >not be conclusive [quoted text clipped - 19 lines] > we can prove the Godel sentence for PA in any more conclusive > fashion than a machine can. That's not what I am getting at. > -- > Daryl McCullough > Ithaca, NY Newberry says... >On Nov 24, 7:12 am, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> A proof is relative to a set of axioms. > >"PA is consistent" means that in PA, for any P we will never derive P >& ~P no matter how long and in what order we keep generating theorems. >It is not relative to any axioms. It either is the case or not. This >is what we are interested in proving. But to "prove" something means to derive it from a set of axioms. Proof is relative to a set of axioms, but *truth* is not. So whether PA is consistent or not is a matter of fact, independent of any axioms. But whether we can *prove* that fact depends on what axioms we are using. >> There are axioms that >> are strong enough to prove the consistency of PA, so your claim [quoted text clipped - 3 lines] > >So how can we possible arrive at the conclusion that PA is consistent? Because it follows from other things we believe. There is no absolute sense in which we know it (or anything else). It's just that the consistency of PA follows from our best theories of mathematics. >So which one is it? >A) We don't prove the correctness of those intuitions >B) it's perfectly well formalizable Both are true. The consistency of PA follows from our intuitions. Our intuitions are pretty much formalizable. But we can't prove the correctness of those intuitions. >We can even try one hypothesis after another but not both at the same >time. [quoted text clipped - 24 lines] > >That's not what I am getting at. Well, I don't know what you are getting at, then. But it is not true that Godel's theorem implies that humans surpass any machine. -- Daryl McCullough Ithaca, NY On Nov 24, 11:25 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 12 lines] > of any axioms. But whether we can *prove* that fact depends on > what axioms we are using. If the fact is independent of any axioms then I wonder what we need the axioms for. Let's forget about proofs then. How do we determine if "PA is consistent" is true? > >> There are axioms that > >> are strong enough to prove the consistency of PA, so your claim [quoted text clipped - 8 lines] > just that the consistency of PA follows from our best theories > of mathematics. Which theories? ZFC? We cannot prove that ZFC is consistent. At best we know it in the same sense we know PA is consistent. So I do not know what follows from what. > >So which one is it? > >A) We don't prove the correctness of those intuitions [quoted text clipped - 3 lines] > intuitions. Our intuitions are pretty much formalizable. > But we can't prove the correctness of those intuitions. By formalization I mean a formal proof. We were discussing the fact that no matter how long we generate theorems of PA we will never derive a contradiction. How did we arrive at the knowledge that it is true? Is the method by which we arrived at it formalizable? (OK, there is a proof in ZFC that an arithmetical statement in some sense equivalent to "PA is consistent" is derivable from its axioms. That is not what we are interested in. We are interested in whether "PA is consistent" is true. How did we arive at the knowledge of this truth? Is the method we arrived at it formalizable?) I think you would agree with me that the answer is A. > >We can even try one hypothesis after another but not both at the same > >time. [quoted text clipped - 34 lines] > > - Show quoted text - > We are interested in whether "PA is > consistent" is true. How did we arive at the knowledge of this truth? > Is the method we arrived at it formalizable? The issue was whether a particular argument that leads to a certain conclusion (the consistency of PA) is formalizable. The question of the "method" by which we hit on the argument in question is quite different. It may be mere luck that we hit on a proof: that doesn't affect the question of whether the proof, when discovered, is susceptible to formalization. > > We are interested in whether "PA is > > consistent" is true. How did we arive at the knowledge of this truth? > > Is the method we arrived at it formalizable? > > The issue was whether a particular argument that leads to a certain > conclusion (the consistency of PA) is formalizable. Is it? > The question of the "method" by which we hit on the argument in > question is quite different. By "method" obviously I did NOT mean the method of arriving at proofs. I used "method" instead of "proof" to avoid certain connotations. > It may be mere luck that we hit on a proof: that doesn't affect the > question of whether the proof, when discovered, is susceptible to > formalization. I do not understand this. Let's first decide if the argument that leads to "No matter how long we generate theorems of PA we will never derive a contradiction" is true is formalizable. In particular we are interested in the argument that makes us certain that this is the case, not the formalization in ZFC. Not knowing if ZFC is consistent we do not know if the ZFC proof produces a truth or a falsehood. > I do not understand this. Let's first decide if the argument that > leads to > "No matter how long we generate theorems of PA we will never > derive a contradiction" is true is formalizable. We are not going to be able to decide that. We are especially not going to be able to watch *PA* itself decide it, since PA cannot and therefore DOES not decide this. Stronger theories can decide it, but they don't so much "decide" it as *presume* it. Every theorem is basically ALREADY ASSUMED by the axioms. Newberry says... >On Nov 24, 11:25 am, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> But to "prove" something means to derive it from a set of axioms. >> Proof is relative to a set of axioms, but *truth* is not. So [quoted text clipped - 4 lines] >If the fact is independent of any axioms then I wonder what we need >the axioms for. The point of axioms is to show that the truth of a complex, not intuitively obvious statement follows from the truth of statements that you've already accepted. >Let's forget about proofs then. How do we determine if >"PA is consistent" is true? You have to look at what the axioms of PA *say*. The Peano axioms are about the natural numbers, which has an intuitive interpretation in terms of counting, pebbles, for instance. The successor operation corresponds to adding another pebble to a collection. If x was the number beforehand, then x+1 is the number afterwards. The addition operation corresponds to combining two collections of pebbles. If x is the number in the first collection, and y is the number in the second collection, then x+y is the number in the combined collection. The multiplication operation corresponds to the operation of lining up pebbles in rectangular arrays. If you have an array of x rows of pebbles, and y pebbles in each row, then you have x*y pebbles altogether. The axioms of PA are all intuitively true about this interpretation. We accept them as true because we can hardly conceive of how they can be false. But it's just a matter of firmness of intuitions. Our intuitions *could* be wrong, I suppose. But if we are wrong about something as simple as arithmetic, then I don't think that there is a single aspect of mathematics that would be salvageable. We might as well treat arithmetic as correct until we have good reason to believe otherwise. >> Because it follows from other things we believe. There is no >> absolute sense in which we know it (or anything else). It's >> just that the consistency of PA follows from our best theories >> of mathematics. > >Which theories? ZFC? We cannot prove that ZFC is consistent. So what? You can only prove things from axioms. Either those axioms are left unproved, or you have an infinite chain of axioms built on top of axioms, with no foundation. Those are the choices. >> >So which one is it? >> >A) We don't prove the correctness of those intuitions [quoted text clipped - 5 lines] > >By formalization I mean a formal proof. That's what I mean, too. There is a formal proof of the consistency of PA. >We were discussing the fact >that no matter how long we generate theorems of PA we will never >derive a contradiction. How did we arrive at the knowledge that it is >true? You could take it to be an empirical fact, tested by centuries of experience. >Is the method by which we arrived at it formalizable? Yes. -- Daryl McCullough Ithaca, NY "Newberry" <newberryxy@gmail.com> ha scritto >> >So which one is it? >> >A) We don't prove the correctness of those intuitions [quoted text clipped - 8 lines] > derive a contradiction. How did we arrive at the knowledge that it is > true? We have 2 methods: 1) We consider the issue from a sintactic point of view and prove it by transfinite induction (Gentzen). 2) We find a model for the axioms. > Is the method by which we arrived at it formalizable? Both 1 & 2 are formalizable in ZFC *BUT* we DON'T know that PA is consistent because we formalized it in ZFC (*), we instead DO know it by mere consideration of the intuitive not-formalized arguments 1 or 2 above. (*) Otherwise you could ask why should ZFC influence our belief given the fact that we don't even know if it is consistent. > 1) We consider the issue from a sintactic point of view and prove it by > transfinite induction (Gentzen). > 2) We find a model for the axioms. ... > Both 1 & 2 are formalizable in ZFC Right. > *BUT* we DON'T know that PA is consistent > because we formalized it in ZFC (*), Of course we do. > we instead DO know it by mere > consideration of the intuitive not-formalized arguments 1 or 2 above. No, really, you don't know much of ANything, especially not anything infinitary, THAT way. > (*) Otherwise you could ask why should ZFC influence our belief given the > fact that we don't even know if it is consistent. True. But THAT is when you say that it looks intuitively right. The point is, once you say this for ZFC, you never have to say it for anything else. "george" <greeneg@cs.unc.edu> ha scritto >> 1) We consider the issue from a sintactic point of view and prove it by >> transfinite induction (Gentzen). [quoted text clipped - 15 lines] > No, really, you don't know much of ANything, especially > not anything infinitary, THAT way. I desagree. The majority of mathematicians usually talk about infinitary things and are convinced about their arguments without even know what ZFC is (and also knowing nothing about first order formal systems). > >> we instead DO know it by mere > >> consideration of the intuitive not-formalized arguments 1 or 2 above. [quoted text clipped - 4 lines] > I desagree. The majority of mathematicians usually talk about infinitary > things and are convinced about their arguments without even know what ZFC is But you said "arguments". "Argument" implies a chain of REASONING, of logical INFERENCE. In other words, there are some axioms and rules of inference being invoked to support these beliefs. It doesn't have to be ZFC specifically. Lots of things will work. ZFC is just the currently usual choice for people who choose to care about this aspect of it. > (and also knowing nothing about first order formal systems). It's not like religion. It works whether you believe in it or not. "george" <greeneg@cs.unc.edu> ha scritto >> >> we instead DO know it by mere >> >> consideration of the intuitive not-formalized arguments 1 or 2 above. [quoted text clipped - 9 lines] > logical INFERENCE. In other words, there are some axioms and rules > of inference being invoked to support these beliefs. Axioms and rules of inference are connected only to *formal* reasoning. Informal reasoning don't imply any defined set of axioms or rules, even if it can if course be formalized in many different formal deductive systems. You should also agree that we definitely don't need a reasoning to be framed in a set of axioms and rules to be convinced. Of course it can help, but we are convinced (for example) of infinity of primes or Pythagorean theorem long before we hear about PA, Q, ZFC or Hilbert's axioms for geometry. >> (and also knowing nothing about first order formal systems). > > It's not like religion. It works whether you believe in it or not. I know. The point is that even if you could be able to formalize the reasoning of mathematicians in ZFC, this formalization is not really relevant for their work and their beliefs. > "george" <gree...@cs.unc.edu> ha scritto > [quoted text clipped - 28 lines] > reasoning of mathematicians in ZFC, this formalization is not really relevant > for their work and their beliefs. The discrepancy is that the formalized proofs of PA's consistency have zero cogency while the intuitive informal arguments - so I am told - have 100% cogency. In this sense the claim tha that the intuitive argument is formalizable is misleading. > The discrepancy is that the formalized proofs of PA's consistency have > zero cogency while the intuitive informal arguments - so I am told - > have 100% cogency. This is exactly the opposite of the truth. Every individual step of any formal argument has ABSOLUTELY MAXIMAL cogency. If you know that P v Q v R etc. and you also know that ~P, then the argument that from these, the shorter Q v R etc. MUST follow, has ABSOLUTELY MAXIMAL cogency. There is a version of FOL in which THIS is THE ONLY inference rule. So all the formalized proofs OF ANYthing that you CAN prove in FOL have maximal cogency. It is the informal ones that may be open to doubt. Of course, all this is meaningless if the axioms are inconsistent. But there is certainly nothing even vaguely resembling a cogent argument that ZFC is inconsistent. > Axioms and rules of inference are connected only to *formal* reasoning. WRONG. *ALL* *reasoning* is rule-based. That's what MAKES it *reasoning*. > Informal reasoning don't imply any defined set of axioms or rules, even if it > can if course be formalized in many different formal deductive systems. EXACTLY. It is the fact that it CAN be formalized that MAKES it REASONING. And in fact, in every individual case, the individual reasoner IN FACT IS USING axioms and rules. The fact that he is not being overtly explicitly conscious of which ones they are -- especially when it doesn't matter -- does NOT imply that they are not there. It does not imply that his reasoning behavior is not factually OBSERVABLY CONFORMING to certain patterns and rules. > You should also agree that we definitely don't need a reasoning to be framed > in a set of axioms and rules to be convinced. That is NOT the point. The point is that IF it IS "reasoning" AT ALL, and if it is "convincing" at all, then it is convincing BECAUSE OF its CONFORMANCE to axioms and rules. > Of course it can help, but we > are convinced (for example) of infinity of primes or Pythagorean theorem long > before we hear about PA, Q, ZFC or Hilbert's axioms for geometry. Perhaps, but the point is, whatEVER argument convinced you of them, it made use of SOME form of axioms, definitions, and rules of inference. The fact that nobody wrote them all down in advance and required you to put all the right pegs in the right holes does NOT imply that you were not, in fact, while working through the reasoning, doing exactly that. > I know. The point is that even if you could be able to formalize the > reasoning of mathematicians in ZFC, this formalization is not really relevant > for their work and their beliefs. Of course, since ANY formulation will do. The point is it is simply BETTER to KNOW what formulation you are using. For a long time I did not know (after FLT was proved) whether FLT followed from PA or not. It turns out it was proved long ago that it doesn't. But the fact that that proof got as much acceptance as it got withOUT it being clear what axioms it was being proved from IS ENTIRELY A BAD thing. "george" <greeneg@cs.unc.edu> ha scritto >> Axioms and rules of inference are connected only to *formal* reasoning. > > WRONG. *ALL* *reasoning* is rule-based. That's what MAKES > it *reasoning*. Maybe, but we don't know which are these axioms and rules in general, so it can be a controversial point of view to assume that they always characterize every kind of reasoning. >> Informal reasoning don't imply any defined set of axioms or rules, even if >> it >> can if course be formalized in many different formal deductive systems. > > EXACTLY. It is the fact that it CAN be formalized that MAKES it > REASONING. Well, consider that the concept of "reasoning" do not belong only to mathematics. We have reasonings in philosophy, psichology, law, politics... most of them don't seem to be so easily "formalizable". Consider also that even 2nd order Peano arithmetic is not formalizable. > And in fact, in every individual case, the individual reasoner IN FACT > IS USING [quoted text clipped - 5 lines] > behavior is not > factually OBSERVABLY CONFORMING to certain patterns and rules. If you allow those "axioms and rules" to be not conscious than I could still be right when I say that: >> we DON'T know that PA is consistent >> because we formalized it in ZFC (*), we instead DO know it by mere >> consideration of the intuitive not-formalized arguments 1 or 2 above. because we could still think that we are convinced by these informal reasonings because of my inconscious rules and axioms. >> You should also agree that we definitely don't need a reasoning to be >> framed [quoted text clipped - 3 lines] > and if it is "convincing" at all, then it is convincing BECAUSE OF its > CONFORMANCE to axioms and rules. Well, I actually don't know why the human mind find things convincing or not, maybe there are some axioms and rules hidden inside it or maybe not. This was not the issue I was addressing however. >> Of course it can help, but we >> are convinced (for example) of infinity of primes or Pythagorean theorem [quoted text clipped - 4 lines] > it made use of SOME form of axioms, definitions, and rules of > inference. Yes, you can of course say that because we know that these arguments are formalizable and you are considering a weak form of "use" of the axioms and rules (it could be inconscious). I obviously can agree with your view, but my point was another: I'm saying that we don't need to actually formalize an argument in ZFC, PA or any other formal system in order to be convinced. >> I know. The point is that even if you could be able to formalize the >> reasoning of mathematicians in ZFC, this formalization is not really [quoted text clipped - 7 lines] > it > doesn't. Fermat's Last Theorem has proven to be undecidable in PA? > But the fact that that proof got as much acceptance as it > got > withOUT it being clear what axioms it was being proved from > IS ENTIRELY A BAD thing. I'm not responsible :) > Consider also that even 2nd order Peano arithmetic is not formalizable. I don't know what you mean by '2nd order Peano arithmetic', but there is a formal theory that is second order Peano arithmetic. MoeBlee "MoeBlee" <jazzmobe@hotmail.com> ha scritto >> Consider also that even 2nd order Peano arithmetic is not formalizable. > > I don't know what you mean by '2nd order Peano arithmetic', but there > is a formal theory that is second order Peano arithmetic. Ok, actually second order *logic* is not formalizable. Consider any formalization L2 that is supposed to represent every possible 2nd order deduction. Consider the formula asserting "(the conjunction of the 2nd order Peano Axioms) -> G" where G is the Godel formula for L2. This formula is true in any 2nd order model, so it represents a correct 2nd order deduction. Yet it is not covered by our formalization. > "MoeBlee" <jazzmobe@hotmail.com> ha scritto > [quoted text clipped - 5 lines] > > Ok, actually second order *logic* is not formalizable. It is formalizable in the same sense that 2nd order PA is (or first order PA, for that matter). There's just no semantically complete, consistent, recursive axiomatization of it (just as there is no negation complete, consistent, recursive axiomatization of PA). > Consider any formalization L2 that is supposed to represent every > possible 2nd order deduction. You mean a formalization in which every 2nd order validity (in the language of the formalization) is provable? > Consider the formula asserting "(the conjunction of the 2nd order > Peano Axioms) -> G" where G is the Godel formula for L2. This formula > is true in any 2nd order model, so it represents a correct 2nd order > deduction. I guess you mean a second-order validity. > Yet it is not covered by our formalization. So, assuming "covered by" means "provable in", it appears that, for you, a formalization has to be complete. "Chris Menzel" <cmenzel@remove-this.tamu.edu> ha scritto >> Ok, actually second order *logic* is not formalizable. > > It is formalizable in the same sense that 2nd order PA is (or first > order PA, for that matter). There's just no semantically complete, > consistent, recursive axiomatization of it (just as there is no negation > complete, consistent, recursive axiomatization of PA). Ok. >> Consider any formalization L2 that is supposed to represent every >> possible 2nd order deduction. > > You mean a formalization in which every 2nd order validity (in the > language of the formalization) is provable? Yes, I was meaning a complete formalization of second order reasoning. Incomplete formalizations would not allow to say that 2nd order reasonings are formalizable in general (see below). >> Consider the formula asserting "(the conjunction of the 2nd order >> Peano Axioms) -> G" where G is the Godel formula for L2. This formula >> is true in any 2nd order model, so it represents a correct 2nd order >> deduction. > > I guess you mean a second-order validity. Yes. >> Yet it is not covered by our formalization. > > So, assuming "covered by" means "provable in", it appears that, for you, > a formalization has to be complete. Yes... The discussion was getting a bit phylsophical. The point was: does every reasoning have to be formalizable in order to be a "reasoning"? Do there exist rules of reasoning that we follow even when we are not conscient of them? In this context a relevant "formalization" of reasonings should be complete, I think, otherwise evrything is trivially formalizable by an "ad hoc" incomplete set of rules. > The point was: does every > reasoning have to be formalizable in order to be a "reasoning"? Do there > exist rules of reasoning that we follow even when we are not conscient of > them? In this context a relevant "formalization" of reasonings should be > complete, I think, otherwise evrything is trivially formalizable by an "ad > hoc" incomplete set of rules. You can advocate such a thesis, but perhaps you should consider doing it with terminology that does not so markedly depart from the ordinary. A system of theory is formal or not depending on whether it has a recursive axiomatization (and recursive rules of inference), and not depending on whether the system or theory proves every formula that is entailed by its rules and axioms. MoeBlee > "MoeBlee" <jazzm...@hotmail.com> ha scritto > [quoted text clipped - 9 lines] > formula is true in any 2nd order model, so it represents a correct 2nd order > deduction. Yet it is not covered by our formalization. I wouldn't put it that way. Rather, I would say: There is a formal system that is called 'second order logic', but (unlike first order logic) with second order logic it is not the case that every validity is provable; and there is a formal system that is called 'second order PA', of which it is not the case that every formula entailed by the axioms is provable from the axioms. What makes a system formal is that it is recursively axiomatized, not whether the system proves every formula that is entailed by the axioms. MoeBlee > For a long time I did not know (after FLT was proved) whether > FLT followed from PA or not. It turns out it was proved long ago that [quoted text clipped - 3 lines] > withOUT it being clear what axioms it was being proved from > IS ENTIRELY A BAD thing. I don't think it has been proved that FLT is independent from the axioms of PA, if that's what you mean. That would be quite some result. Do you mean that it has been shown that the proof of FLT as given by Wiles cannot be formalised in PA? That would not be surprising as the proof uses a lot of analysis of modular functions. On Dec 3, 7:00 pm, kleptomaniac6...@hotmail.com wrote: > I don't think it has been proved that FLT is independent from the > axioms of PA, if that's what you mean. I do. It definitely has. It is actually an old result, like over 40 years. [27] Shepherdson, John C., A non-standard model for a free variable fragment of number theory, Bulletin de l'Academie Polonaise des Sciences. Serie des Sciences Mathematiques, Astronomiques et Physiques, 12, 1964, 79-86. > That would be quite some result. > Do you mean that it has been shown that the proof of FLT as > given by Wiles cannot be formalised in PA? Well, indirectly. It used to be in vogue to construct nonstandard models of PA. Back when it was (in the early '60s), Shepherdson constructed one in which FLT failed for the *first*/simplest case (cubic). This doesn't have any direct relevance to the TRW proof, but it does imply independence from PA. I'm almost sure (though my whole point is, this hasn't been made *clearly* explicit) it's not independent of ZFC. >On Dec 3, 7:00 pm, kleptomaniac6...@hotmail.com wrote: >> I don't think it has been proved that FLT is independent from the >> axioms of PA, if that's what you mean. > >I do. It definitely has. It is actually an old result, like over 40 >years. No, Shepherdson isn't studying PA in that paper. It is still open whether FLT is independent of PA. SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences On Dec 4, 8:21 pm, tc...@lsa.umich.edu wrote: > No, Shepherdson isn't studying PA in that paper. It is still open whether > FLT is independent of PA. OK. I was reading too fast. From http://web.mat.bham.ac.uk/R.W.Kaye/papers/tennenbaum/tennhistory which includes this paragraph -- > To my mind, the highlight of this period of building recursive models for > the purposes of independence results was the results of the early 1960s > by Shepherdson, who, using algebraic methods, produced beautiful > nonstandard models of quantifier-free arithmetic in which he showed > number theoretic results such as the infinitude of primes and Fermat's > Last Theorem (in fact, for exponent 3) are false (MR0161798). Apparently(in hindsight),this was a recursive model of a weaker(than PA)theory. What was (is) needed is a non-recursive/non- standard model of PA. > On Dec 3, 7:00 pm, kleptomaniac6...@hotmail.com wrote: > [quoted text clipped - 21 lines] > point is, > this hasn't been made *clearly* explicit) it's not independent of ZFC. I thought Harvey Friedman suspects that FLT can be proved in PA? Also, would it not be the case that independence from PA would imply that FLT is true? As FLT can be expressed as a pi-1 sentence. On Dec 4, 8:32 pm, kleptomaniac6...@hotmail.com wrote: > I thought Harvey Friedman suspects that FLT can be proved in PA? OK, I was wrong. I was reading this (too fast) from a paper by Kaye on Tennenbaum's theorem: > As for independence results, Kemeny did produce a model > that addressed some questions of independence in 1958 (MR0098685). [quoted text clipped - 7 lines] > and Fermat's Last Theorem (in fact, for exponent 3) are false > (MR0161798) [that was the reference I cited] This was apparentely a nonstandard model of "quantifier-free arithmetic", whatever that is. Presumably, after you put the quantifiers and a full first- order induction axiom back in, this model is not adequate as a model of PA. But I didn't see that the first time. Apparently this model was also recursive, which of course a nonstandard model of PA could not be. > By this time Tennenbaum's theorem was already well known, > and Shepherdson and others were certainly aware of the limitations > of this approach. Limitations apparently preventing any such earth-shaking indepenence results as I had been expecting. > Also, would it not be the case that independence from PA would imply that > FLT is true? > As FLT can be expressed as a pi-1 sentence. I do not see how to do that (express FLT as a Pi-1 sentence in the language of PA), since the language of PA does not include an exponentiation operator. > On Dec 4, 8:32 pm, kleptomaniac6...@hotmail.com wrote: >> Also, would it not be the case that independence from PA would imply that >> FLT is true? [quoted text clipped - 3 lines] > language of PA), since the language of PA does not include an > exponentiation operator. the expressibility of exponentiation in PA was done by Goedel in his proof of incompleteness (and it's serious work). Not sure if this would allow a pi-1 formulation, though. SignatureAlan Smaill > > On Dec 4, 8:32 pm, kleptomaniac6...@hotmail.com wrote: > >> Also, would it not be the case that independence from PA would imply that [quoted text clipped - 11 lines] > -- > Alan Smaill I thought (though I could be wrong) that there was a general principle that any statement in the language of first order arithmetic which took the form: every positive integer has the property p, where p is a property that can be algorithmically checked can be expressed as a pi-1 sentence. In the case of FLT, if every positive integer n had the property that it was not the z in a counterexample to FLT, so to speak. To check that a number n is not involved as z in a counterexample to FLT, all that is required is that one check if n is a perfect power where the power is higher than 2, and then check all the pairs of perfect powers which are less than z, of which there will be finitely many. Obviously, propositions like "this diophantine equation has finitely many solutions" would not take this form. It cannot be shown false by counterexample. >> > On Dec 4, 8:32 pm, kleptomaniac6...@hotmail.com wrote: >> >> Also, would it not be the case that independence from PA would imply that [quoted text clipped - 17 lines] > property that can be algorithmically checked can be expressed as a > pi-1 sentence. I'm not convinced; according to Smullyan, the relation x^y = z is sigma-1 wrt PA. >In the case of FLT, if every positive integer n had the > property that it was not the z in a counterexample to FLT, so to [quoted text clipped - 7 lines] > many solutions" would not take this form. It cannot be shown false by > counterexample. SignatureAlan Smaill > >> > On Dec 4, 8:32 pm, kleptomaniac6...@hotmail.com wrote: > >> >> Also, would it not be the case that independence from PA would imply that > >> >> FLT is true? [quoted text clipped - 19 lines] > I'm not convinced; > according to Smullyan, the relation x^y = z is sigma-1 wrt PA. In "Godel, Escher, Bach" Hofsteder mentions the difficulty of defining "z is a power of x" in PA and challenges the reader to define the simpler "z is a power of two". I came up with "z has no odd factor greater than one". I don't know if that's pi-1 or sigma-1. :-) -- hz On Dec 5, 5:27 pm, kleptomaniac6...@hotmail.com wrote: > I thought (though I could be wrong) that there was a general principle > that any statement in the language of first order arithmetic which > took the form: every positive integer has the property p, where p is a > property that can be algorithmically checked can be expressed as a > pi-1 sentence. We can concede that. In the case of FLT, though, that will not help. You cannot algorithmically check the property because it has to apply to infinitely many triples. You can algorithmically refute it if it's false but you cannot algorithmically confirm it if it's true. > In the case of FLT, if every positive integer n had the > property that it was not the z in a counterexample to FLT, so to > speak. No, NOT *so* to speak. That is incorrect. The correct way of speaking it, assuming you have an exponentiation operator, is An[n>2-->Axyz[~x^n+y^n=z^n] ]. IT IS *n* that must have or lack the property. You have to check the n. That means you have to check infinitely many x,y,z's. That is not algorithmically doable (yet). If anyone actually finds the algorithm then FLT will get proved from PA rather quickly. > To check that a number n is not involved as z is irrelevant; n doesn't need to be involved as z; n is involved as n. > On Dec 5, 5:27 pm, kleptomaniac6...@hotmail.com wrote: > > In the case of FLT, if every positive integer n had the > > property that it was not the z in a counterexample to FLT, so to > > speak. > No, NOT *so* to speak. That is incorrect. > The correct way of speaking it, assuming you have an exponentiation [quoted text clipped - 5 lines] > actually finds the algorithm then FLT will get proved from PA rather > quickly. But you can eliminate the inner quantifiers if you use tuples: every integer has the property that { if you decode a quadruple <x,y,z,n> from it, then n<3 or x^n + y^n != z^n } Signaturepa at panix dot com > > On Dec 5, 5:27 pm, kleptomaniac6...@hotmail.com wrote: > > > In the case of FLT, if every positive integer n had the > > > property that it was not the z in a counterexample to FLT, so to > > > speak. I replied: > > No, NOT *so* to speak. That is incorrect. > > The correct way of speaking it, assuming you have an exponentiation > > operator, is > > An[n>2-->Axyz[~x^n+y^n=z^n] ]. IT IS *n* that must have or lack the > > property. You have to check the n. > But you can eliminate the inner quantifiers if you use tuples: > > every integer has the property that { > if you decode a quadruple <x,y,z,n> from it, > then > } But that is missing the point,too. You could just say Axyzn[n<3 or x^n + y^n != z^n]. Whether that is or isn't Pi-1 is, precisely as you say, not even worth arguing about because you can encode the tuple. The fact that that is possible is a good reason for writing it the way I write it (with 1 quantifier symbol instead of 4). It also bears stressing that this is not even a tuple, that the order DOES NOT matter, that any of the other 23 orders of the 4 variables would be THE SAME sentence. What DOES matter and what IS the point is that a "cuonterexample to FLT" is going to be THE *n* and NOT the "z", as klept0 was mis- alleging. And it also matters that you have to have some sort of Pi-1 definition of exponentiation to get this into the language of PA. I presume the original Godel proof uses the chinese remainder theorem among other things to manage that. > What DOES matter and what IS the point is that a "cuonterexample > to FLT" is going to be THE *n* and NOT the "z", as klept0 was mis- [quoted text clipped - 6 lines] > Godel proof uses the chinese remainder theorem among other things to > manage that. Yes it does. SignatureAlan Smaill > On Dec 5, 5:27 pm, kleptomaniac6...@hotmail.com wrote: > [quoted text clipped - 29 lines] > n doesn't need to be involved as z; > n is involved as n. Pick a number z. There is an algorithm which tests whether z takes place as the largest power in a counterexample to FLT. First you check whether z is a perfect power that is not a square. If not, you know z cannot take part in a counterexample. If z is a perfect power, it will be a perfect power in x ways, i.e. with x different exponents. For each exponent n, check all the pairs of nth powers both of which which are less than z, and see whether they add up to z. If for each n no such pair of nth powers is found by this finite search, then z does not take part in a counterexample to FLT. FLT is equivalent to the assertion that for every number z, the algorithm I just described will return negative. > "PA is consistent" means that in PA, for any P we will never derive P > & ~P no matter how long and in what order > we keep generating theorems. Exactly so. But the question is, WHAT axioms are you trying to prove this FROM? If you are trying to prove it from PA itself, then it is hopeless (unless PA is inconsistent, in which case it will allow you to prove both that it is consistent AND THAT IT ISN'T). > It is not relative to any axioms. This is entirely wrong. Whether something IS (or is not) a *valid* or *sound* STEP in a proof, and whether some stringing- together of such steps is or is not a proof, IS axiomatically definable, at least up to finitude. Whether something does or does not CONSTITUTE a proof (or proof-subtree) IS relative to axioms. There are DIFFERENT possible systems of logic and rules of inference. You DO NEED to specify the one you are using -- the one that defines "proof", as you define it -- axiomatically. In the first-order case this is sort of fundamentally guaranteed to fail because we intuitively require that proofs must be finite, and finitude is not first-order definable. But you need to clarify your other confusions before falling on *that* sword. > It either is the case or not. True, but which of these is the case *does* depend on the *meta*-theoretical axioms that you used in *your* logic's definition of "proof". > This is what we are interested in proving. Who "we", dumbass? The whole import of Godel's theorem is that this CAN'T be proved. "Newberry" <newberryxy@gmail.com> ha scritto > There is no problem >> having a truth predicate for arithmetic in a language that [quoted text clipped - 4 lines] > somehow inherently impossible or it "can't" be done because a > contradiction would result? The reason is that if P is a theory including an axiom saying "(P proves S) -> S" then the statement "(P proves S) -> S" is *provably false*. "LordBeotian" <pokipsy76@yahoo.it> ha scritto >> There is no problem >>> having a truth predicate for arithmetic in a language that [quoted text clipped - 7 lines] > The reason is that if P is a theory including an axiom saying "(P proves > S) -> S" then the statement "(P proves S) -> S" is *provably false*. Ops, I think I have answered to another question... :| "Newberry" <newberryxy@gmail.com> ha scritto > There is no problem >> having a truth predicate for arithmetic in a language that [quoted text clipped - 4 lines] > somehow inherently impossible or it "can't" be done because a > contradiction would result? 1) Truth, despite provability, is not a sintactic property. There is no reason to expect that there exist an algorithm that will say if a number is the godel number of a true arithmetical wff. 2) Actually the assumption that such an algoritm exist lead to a contraddiction. "Newberry" <newberryxy@gmail.com> ha scritto >> >So let's confine ourselves to PA for now. We can prove that it is >> >consistent, that is we have proven G. How did we manage to do that [quoted text clipped - 7 lines] > Right. So it means that Lucas and Penrose are rigtht, and Franzen is > wrong? How would you draw this conclusion? > On Nov 9, 4:17 am, aatu.koskensi...@xortec.fi wrote: >> Yes, PA is obviously consistent. > > OK, how do we reconcile it with this? Reconcile in what sense? There is no apparent contradiction between Torkel's explanation concerning... ... the mistaken idea that "Gödel's theorem states that in any consistent system which is strong enough to produce simple arithmetic there are formulas which cannot be proved in the system, but which we can see to be true." The theorem states no such thing. As has been emphasized, in general we simply have no idea whether or not the Gödel sentence of a system is true, even in those cases when it is in fact true. What we know is that the Gödel sentence is true if and only if the system is consistent, and that much is provable in the system itself. and the observation that PA is obviously consistent. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus On Nov 12, 3:49 am, Aatu Koskensilta <aatu.koskensi...@xortec.fi> wrote: > > On Nov 9, 4:17 am, aatu.koskensi...@xortec.fi wrote: > >> Yes, PA is obviously consistent. [quoted text clipped - 14 lines] > > and the observation that PA is obviously consistent. There are several issues here. 1) Isn't exhibiting one theory (PA/ZFC) good enough to establish Lucas's argument? 2) What did TF intend to say by "in general"? Did he mean a) the meta, meta-theories in which we establish the consistency of PA and then ZFC etc. Or did he mean b) alternative theories e.g. Quine's set theory The problem in a) is that there seems to be an infinite regress. As far as b) chances are that we will be able to establish their consistency just like we established the consistency of PA/ZFC. > 1) Isn't exhibiting one theory (PA/ZFC) good enough to establish > Lucas's argument? No. > 2) What did TF intend to say by "in general"? He means that in general, if we're presented with an axiomatisable extension of Q we quite literally have no idea whether or not it is consistent, and consequently whether or not its Gödel sentence is true or not. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus > On Nov 12, 3:49 am, Aatu Koskensilta <aatu.koskensi...@xortec.fi> > wrote: [quoted text clipped - 32 lines] > > - Mostrar texto de la cita - If I don't misunderstand your query on infinite regress along the hierarchy of theories, you are posing an 'ultimately philosophical' question: where does our confidence in PA ultimately stems from? Well, it originates from our confidence in reason, in rational evidence. That is what Lotze called 'Selbstvertrauen der Vernunft', i.e. reason's confidence in reason. We believe some propositions because we are able to derive them from evident truths. We believe evident truths because we rely on reason. We rely on reason for no reason? Regards > We rely on reason for no reason? Relying on reason, and accepting evident truths, is very reasonable. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus On Nov 13, 1:02 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi> wrote: > Relying on reason, and accepting evident truths, is very reasonable. I completely agree. But, is that a good reason to rely on reason? Regards > > On Nov 12, 3:49 am, Aatu Koskensilta <aatu.koskensi...@xortec.fi> > > wrote: [quoted text clipped - 35 lines] > If I don't misunderstand your query on infinite regress along the > hierarchy of theories, I was mainly asking if I interpreted Franzen correctly. We are sure that PA is consistent and we prove it in ZFC. We are sure that ZFC is consistent and we prove it in some metatheory. But we are not sure if this meatatheory is consistent. Is this what he is saying? you are posing an 'ultimately philosophical' > question: where does our confidence in PA ultimately stems from? > [quoted text clipped - 10 lines] > > - Show quoted text - > I was mainly asking if I interpreted Franzen correctly. We are sure > that PA is consistent and we prove it in ZFC. We are sure that ZFC is > consistent and we prove it in some metatheory. But we are not sure if > this meatatheory is consistent. Is this what he is saying? I wouldn't take that as an accurate summary of his view. Rather, he has a main point in his discussion of skepticism. If you go back to read it, it's really difficult to miss what that point is. MoeBlee > > > On Nov 12, 3:49 am, Aatu Koskensilta <aatu.koskensi...@xortec.fi> > > > wrote: [quoted text clipped - 57 lines] > > > - Show quoted text - When one says one has proved a theorem "for sure", it means one has proved it from axioms that one is "sure" are true. Consistency does not enter the picture. > > > On Nov 12, 3:49 am, Aatu Koskensilta <aatu.koskensi...@xortec.fi> > > > wrote: [quoted text clipped - 40 lines] > consistent and we prove it in some metatheory. But we are not sure if > this meatatheory is consistent. Is this what he is saying? Not, exactly. I don't think he contends the ONLY way we have to convince ourselves of PA's consistency is by means of ZFC or any other theory. I think he means PA reveals itself evidently consistent after just a careful examination of its axioms, that it is a mere question of intuitive evidence. So, I don't think you can take Franzen's position into an indefiniteley delayed skepticism, so to say. Regards > > > > On Nov 12, 3:49 am, Aatu Koskensilta <aatu.koskensi...@xortec.fi> > > > > wrote: [quoted text clipped - 48 lines] > careful examination of its axioms, that it is a mere question of > intuitive evidence. That is exactly what I meant. So there is no conclusive formal proof of PA's consistency. There is an intuitive proof, which is just as compelling as a formal proof. This intuitive proof is not formalizable. It seems to follow that the human mind surpasses any machine. So I am not exactly sure why he is claiming otherwise. He makes a big deal out of G <--> Con(T) and keeps repeating that we cannot conclude about a given theory that its senetnce G is true; we can merely say that G is true if T is consistent. But then he ends up by saying that he is convinced of PA's consistency more than anyone. You can say that you can construct a machine that will prove T(G) - just program the machine with ZFC. But I do not know if it is equivalent to the human mind. For one we do not know if the machine will start producing falsehoods since we do not know if ZFC is consistent. Secondly we are sure of PA's consistency based on the intutive proof. But we are not sure of it consistency based on the ZFC proof. So it seems we are not the machine programmed with ZFC. BTW, how would you formalize the self-evident "truth cannot be inconsistent with itself"? > So, I don't think you can take Franzen's position into an > indefiniteley delayed skepticism, so to say. > > Regards- Hide quoted text - > > - Show quoted text - > As it happens I think there are conclusive reasons > to believe PA consistent. And: On Nov 9, 1:17 pm, aatu.koskensi...@xortec.fi wrote: > Yes, PA is obviously consistent. Conclusive! Obvious! Who could doubt what one learned as a young boy in Sunday school? AK > > > > Yes, PA is obviously consistent. > Conclusive! Obvious! Who could doubt what one learned as a young boy > in Sunday school? I rated that 5 stars. Which I really don't like having to do for people with whom I've had bitter arguments. > Conclusive! Obvious! Who could doubt what one learned as a young boy > in Sunday school? I don't know. What does one learn about Peano arithmetic as a young boy in Sunday school? SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus Aatu Koskensilta says... >> Conclusive! Obvious! Who could doubt what one learned as a young boy >> in Sunday school? > >I don't know. What does one learn about Peano arithmetic as a young boy in >Sunday school? I'm not sure. But an acquaintance of mine explained how the natural numbers can be represented using lambda calculus. He told me he learned it in Church. -- Daryl McCullough Ithaca, NY > >> Conclusive! Obvious! Who could doubt what one learned as a young boy > >> in Sunday school? [quoted text clipped - 5 lines] > numbers can be represented using lambda calculus. He told me he > learned it in Church. And don't forget that Kleeneness is next to Godelness! ------------------------------------------------------- Bill Taylor W.Taylor@math.canterbury.ac.nz ------------------------------------------------------- And God said Let there be numbers And there *were* numbers. Odd and even created he them, He said to them be fruitful and multiply And he commanded them to keep the laws of induction. ------------------------------------------------------- well, i think the point which Franzen emphasize by the truth which we know is, "if PA is consistent, G is true", what we don't know is, directly " G is true" whether we do not have any idea about the consistency of PA, and because the sentence "if PA is consistent, G is true" is also provable by PA, while G is not if it is consistent. Peter_Smith yazdı: > > > > > > > In "G�del's theorem" Torkel Franzen disputes that the theorem > > > > > > > indicates that the human mind surpasses any computer. [quoted text clipped - 33 lines] > conclusive reason". As it happens I think there are conclusive reasons > to believe PA consistent. > > In "Gödel's theorem" Torkel Franzen disputes that the theorem > > indicates that the human mind surpasses any computer. [quoted text clipped - 14 lines] > doesn't refute it -- a point evidently consistent with the first > quote. TF is not saying that Gödel's theorem does not contradict the view that the system is consistent. He says it does not contradict the view that there is no doubt. So who is the one that does not have any doubts? > The third quote you give starts with an emphasized "If" in TF. It is a > triviality (any set of truths is consistent!). > > He is not, at least in those quotations, saying any of (a) to (g). > > > In "Gödel's theorem" Torkel Franzen disputes that the theorem > > > indicates that the human mind surpasses any computer. [quoted text clipped - 28 lines] > > - Show quoted text - Consistency of the system at hand is just another arithmetical statement, like "every prime of the form 4k+1 is a sum of two squares" or "the sum of the divisors of the nth positive integer is less than or equal to Hn + exp(Hn)log(Hn) where Hn is the nth harmonic number". As far as I am aware, the epistemological issues for determining the truth of the consistency of the theory are no different from the issues for those statements I just mentioned. As for doubts, doubt and certainty are human emotions. One could be certain of con(PA) if one could prove that theorem from a list of arithmetical axioms which one felt certain were true. The same as for any other theorem. On Nov 8, 3:30 pm, kleptomaniac6...@hotmail.com wrote: > Consistency of the system at hand is just another arithmetical > statement, like "every prime of the form 4k+1 is a sum of two squares" > or "the sum of the divisors of the nth positive integer is less than > or equal to Hn + exp(Hn)log(Hn) where Hn is the nth harmonic number". No, it is NOT just like THOSE. THOSE are THEOREMS. THOSE are PROVABLE from the axioms of PA and therefore true in all models of PA. The consistency statement for PA is not provable from/in PA. > On Nov 8, 3:30 pm, kleptomaniac6...@hotmail.com wrote: > [quoted text clipped - 7 lines] > true in all models of PA. The consistency statement for PA > is not provable from/in PA. Woah. OK con(PA) is different in that it can be proven in PA. But be careful, the second of those two statements is (equivalent to) the RIEMANN HYPOTHESIS, and it is an important unsolved problem in mathematics. We don't know if it is provable in PA or not (wow, usenet bickering is so much fun!). Actually I wasn't specifically talking about con(PA), even if it may have seemed like it. What I was trying to say was that consistency statements in general, though they are intimately related to Godel's theorem, have no particular epistemological relevance compared to other arithmetical statements. As for PA, the fact that con(PA) is not derivable from the axioms of PA is interesting, but it has no different epistemological status to all the other statements which are not provable in PA. On Nov 8, 3:30 pm, kleptomaniac6...@hotmail.com wrote: > One could be > certain of con(PA) if one could prove that theorem from a list of > arithmetical axioms which one felt certain were true. The same as for > any other theorem. Not really. The whole point about theorems is that truth doesn't even MATTER for them. Even if the axioms are false, the theorems ARE STILL provable from them. > On Nov 8, 3:30 pm, kleptomaniac6...@hotmail.com wrote: > [quoted text clipped - 6 lines] > doesn't even MATTER for them. Even if the axioms are false, > the theorems ARE STILL provable from them. How could you possibly disagree with that statement? Maybe I could substitute "the same as for being certain of the truth of any other theorem" for "The same as for any other theorem." Happy now? > > > >> ... there is no doubt whatever about the consistency of any of the formal systems we use in mathematics. << p. 105 [Newberry's quote of Franzen] > > Your second quote is misleading: What TF in fact wrote was "Nothing in > > Gödel's theorem in any way contradicts the view that there is no doubt [quoted text clipped - 8 lines] > that there is no doubt. So who is the one that does not have any > doubts? To say "there are no doubts" may be understood as a casual way of saying "there is no reasonable basis for doubt" as opposed to an unnecessarily extremely literalistic interpretation that there does not exist in ceratin people the psychological experience of doubt about the constinency of certain formal theories. Franzen's book is written at a very informal level and it is grossly missing the point to split hairs about a non-technical use of such expressions as "there is no doubt", just as when in, say, a debate, someone says, "So there is no doubt whatever that the proposed amendment is too costly", it is not meant literally that there are not people who experience doubt whether the the amdendment is too costly. And, then, with that more reasonable sense ('a reasonable basis for doubt' as opposed to a sweeping claim as to what psychological experiences people have), still Franzen in that particular passage did not say that there are not reasonable bases for doubt but rather that the incompleteness theorem itself does not provide any such reasonable bases. MoeBlee > > In "Gödel's theorem" Torkel Franzen disputes that the theorem > > indicates that the human mind surpasses any computer. [quoted text clipped - 19 lines] > > He is not, at least in those quotations, saying any of (a) to (g). Here is another quote from TF: >> We do of course know the Gödel sentence of, for example PA, to be true since we know PA to be consistent. << p. 117 TF does make the point that the incompleteness theorem does not contradict the view that we know PA/ZFC to be be consistent with absolute certainty. But he also does endorse the view that we know PA to be consistent with absolute certainty (since the axioms are manifestly true.) He clearly believes that the we know PA/ZFC to be consistent with the same certainty as any mathematical theorem i.e. we can prove G. So the question arises how we can construct a machine that can do the same. obviously not by emulating PA/ZFC. > He clearly believes that the we know PA/ZFC There is no such thing as PA/ZFC. PA is one thing. ZFC is another. > to be consistent with the > same certainty as any mathematical theorem The theorem that PA is consistent is provable in ZFC. It is NOT provable from/in PA. That ZFC is consistent has never been proven. It is certainly not provable from ZFC. Bothering to try to prove it in anything stronger is pointless; you would just have the same question about whether that stronger theory was vs. wasn't consistent. > i.e. we can prove G. "G" is NOT *one* thing. There is a DIFFERENT G for every (sufficiently rich) recursive axiom-set. You canNOT prove G(PA) in PA. You canNOT prove G(ZFC) in ZFC. > So the question arises how we can construct a machine that can do the same. > obviously not by emulating PA/ZFC. First-order logic is complete. It has inference rules. You just construct a machine that applies the inference rules repeatedly. There are some treatments with as few as one rule. And there *is* a way of doing this that you could think of as "emulating ZFC". Or emulating anything else for that matter. The point being that the axioms from which you are going to derive this theory are just one more INPUT to the machine. Newberry says... >He clearly believes that the we know PA/ZFC to be consistent with the >same certainty as any mathematical theorem i.e. we can prove G. So the >question arises how we can construct a machine that can do the same. >obviously not by emulating PA/ZFC. Well, here's an attempt at describing an informal metatheory that captures a lot of human metatheoretic reasoning: 1. Every axiom of ZFC is true. 2. For every statement Phi in the language of ZFC, Phi <-> Phi is true. 3. If T is any theory in the language of ZFC, and every axiom of T is true, then every theorem of T is true. This informal theory can prove Con(ZFC) and Con(ZFC + Con(ZFC)), etc. And it's all perfectly mechanical; you can write a program to work out all the consequences of rules 1-3. Of course, we can give a name to this new theory: Let ZFC_1 = the collection of all statements in the language of ZFC that follow from rules 1-3. Then we can come up with yet another theory by modifying rule1: 1'. Every axiom of ZFC_1 is true. Then we could let ZFC_2 be the set of all consequences of rules 1', 2, and 3. etc. -- Daryl McCullough Ithaca, NY On Nov 9, 10:47 pm, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 23 lines] > Let ZFC_1 = the collection of all statements in > the language of ZFC that follow from rules 1-3. So defined, ZFC_1 is not the informal theory you described, since there is no predicate in the language of ZFC expressing the truth predicate for ZFC sentences, by Tarski's indefinability theorem. Regards LauLuna says... >On Nov 9, 10:47 pm, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> 1. Every axiom of ZFC is true. >> [quoted text clipped - 17 lines] >there is no predicate in the language of ZFC expressing the truth >predicate for ZFC sentences, by Tarski's indefinability theorem. That's why I said "all statements in the language of ZFC" rather than "all statements". -- Daryl McCullough Ithaca, NY > Newberry says... > [quoted text clipped - 31 lines] > Then we could let ZFC_2 be the set of all consequences > of rules 1', 2, and 3. etc. Are ZFC_1, ZFC_2 etc. consistent? How do we know that they are? Can a machine generate theories ZFC_1, ZFC_2 etc? Newberry says... >> 1. Every axiom of ZFC is true. >> [quoted text clipped - 23 lines] > >Are ZFC_1, ZFC_2 etc. consistent? They are consistent if ZFC is true. >How do we know that they are? I can't say for sure I do know that they are, but some people might. >Can a machine generate theories ZFC_1, ZFC_2 etc? Sure. -- Daryl McCullough Ithaca, NY "Daryl McCullough" <stevendaryl3016@yahoo.com> ha scritto > Well, here's an attempt at describing an informal metatheory that > captures a lot of human metatheoretic reasoning: [quoted text clipped - 11 lines] > mechanical; you can write a program to work out > all the consequences of rules 1-3. What does it mean "etc." here? LordBeotian says... >"Daryl McCullough" <stevendaryl3016@yahoo.com> ha scritto > [quoted text clipped - 15 lines] > >What does it mean "etc." here? Sorry, I thought it was obvious. We can define a sequence of theories T_n as follows: T_0 = ZFC T_{n+1} = that theory whose axioms consist of all the axioms of T(n) plus the additional axiom Con(T(n)) Then the informal theory described can prove Con(T_n) for every n. -- Daryl McCullough Ithaca, NY > I am not sure that I understand what Franzen is saying. Don't panic; neither did he. > Is he saying that No. > a) We are absolutely certain about the truths of PA, There is no such thing as a truth of PA. PA is an axiom-set. You prove things from it. There are THEOREMS of PA, things that are PROVABLE from PA. Everything else IS FALSE in AT LEAST ONE model of PA, so there is no point in calling it a truth "of PA". > even those PA cannot prove If PA cannot prove it, then there is a model of PA in which it is false, so it is not a "truth of PA". THEORIES *don't have* "truths". "Truth" comes from MODELS. THEORIES have THEOREMS. > b) The consistency of PA can be proven in ZFC Well, this is true, regardless of whether he meant it. But even there, you have to use epsilon_0 induction. > c) Therefore we can write a computer program emulating ZFC that can > generate the truths of PA No, this is false. Just because PA is consistent does NOT mean there is a computer program that can "generate" all its "truths", especially since there aren't any. There is a program that can recursively enumerate all of PA's THEOREMS, yes, but you don't need as much as ZFC to do *that*. That program is not that complicated (unless you want it to be efficient). > d) We are absolutely certain about the truths of ZFC, even those ZFC > cannot prove Again, ZFC, like PA, IS A THEORY, so it does NOT HAVE "truths". > e) There is a theory X in which we can prove the consistency of ZFC Trivially, X=the-theory-whose-only-axiom-is-"ZFC-is-consistent". Torkel Franzen is not saying any of this (he knows better). Your paraphrases are confused. >> I am not sure that I understand what Franzen is saying. > > Don't panic; neither did he. Given that the quoted passage is perfectly clear it seems you're suggesting Franzén managed to write something eminently comprehensible without himself understanding any of it. This is a curious suggestion -- perhaps you have something more sensible in mind? SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus On Nov 12, 6:37 am, Aatu Koskensilta <aatu.koskensi...@xortec.fi> wrote: > Given that the quoted passage is perfectly clear No, it isn't. Gee, that was easy. Even talking about the clarity of *a* passage in the context of an overall take on a global issue is silly to begin with. The meaning and the clarity of the passage have obvious prior dependencies on the coherence of the context. The quoted piece involved 2 different statements on p.105 and a longer prior one from p.55. That triple does NOT fall under the definition of "the quoted passage". "Passages", instead. And once there are 3 of them then they get to have 3 different degrees of clarity. Not to mention relevance. The mere fact that the passages purport to talk about truth at all is unfortunate. That is necessary if you are trying to debunk other people's misconceptions but there is an underlying hubris here, an underlying claim to gnosis, that is far more objectionable for being SIMPLY IRRELEVANT than it is for being conceited. > it seems you're suggesting > Franz?n managed to write something eminently comprehensible without himself > understanding any of it. "True" in natural language is complicated. The mere existence of "this sentence is NOT true" proves THAT. Choosing to talk about some of this stuff as though it were straightforward is part of the disease, not of the cure. > This is a curious suggestion -- perhaps you have > something more sensible in mind? You're being entirely too charitable. Just the usual cantankerous nihilism. > On Nov 12, 6:37 am, Aatu Koskensilta <aatu.koskensi...@xortec.fi> > wrote: >> Given that the quoted passage is perfectly clear > > No, it isn't. What do you find unclear in any of the passages quoted? From your comments below any unclarity seems to stem from your idiosyncratic doubts about the notion of truth when applied to arithmetical statements (even of restricted logical complexity). General worries and conundrums about truth, such as the liar, are irrelevant to such applications. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus > In "Gödel's theorem" Torkel Franzen disputes that the theorem > indicates that the human mind surpasses any computer. [quoted text clipped - 8 lines] > a) We are absolutely certain about the truths of PA, even those PA > cannot prove What do you mean by "the truths of PA"? There are theorems of PA, and they are true in any model in which the axioms of PA are true. And, for each model for the langauge of PA, there is the set of truths in that model; specifically, there is the set of truths in the standard model for the language of first order PA. > b) The consistency of PA can be proven in ZFC ZFC proves first order PA is consistent. I would think Franzen agrees. > c) Therefore we can write a computer program emulating ZFC that can > generate the truths of PA What is a "program emulating ZFC"? And, again, you say, "truths of PA". If you mean the theorems of first order PA, and if by "generate" you mean recursively enumerate, then, yes, there is a recursive enumeration of the theorems of first order PA. If you mean the sentences true in the standard model of the language of first order PA, then it follows from the incompleteness theorem that there is no recursive enumeration of the set of sentences true in the standard model of the language of first order PA. > d) We are absolutely certain about the truths of ZFC, even those ZFC > cannot prove What do you mean by a "truth of ZFC"? If you mean the theorems of ZFC, then note that "a theorem of ZFC that cannot be proven in ZFC" is an oxymoron. (And there is no sentence S of any language such that there is no theory that proves S.) So only you can say what you mean by "the truths of ZFC". > e) There is a theory X in which we can prove the consistency of ZFC For any theory T (even an inconsistent T) there exists theories that prove the consistency of T. That's trivial. "Theory X proves the consistency of theory T" is not necessarily a very "substantive" claim. > f) Therefore we can write a computer program emulating X that can > generate the truths of ZFC Again, what are "the truths of ZFC"? However, since ZFC is a recursively axiomatized theory, there is a recursive enumeration of the theorems of ZFC. > g) We are not certain about the truths of X > ?? Again, for a theory X, what do you mean by "the truths of X"? MoeBlee In his book on GIT, the late TF (to his great credit) challenges > "... the mistaken idea that "Gödel's theorem states that > in any consistent system which is strong enough to > produce simple arithmetic there are formulas which > cannot be proved in the system, but which we can see to be true." > The theorem states no such thing. This forces us to ask, "Does Prof.Peter Smith know Prof.Daniel Isaacson?" Because Isaacson, TEACHING THIS STUFF RIGHT NOW ("Michaelmas term, 2007), flaunts his ignorance of this Torkelian perspective, lecturing, >> "This independence theorem was unprecedented. In the previous >> hundred years the independence of Euclid's fifth postulate had [quoted text clipped - 7 lines] >>>>> The Godel sentence is demonstrably true, >>>>> though not demonstrable in the system for which it is constructed. Jeezus. This person is a professor. I failed to get my Ph.D. at UNC and have been completely unemployable since. I just got fired from what will probably be my last computer job after less than 5 months. My philosophy degree is a bachelor's. But I know that what this professor has just said is utter bullshit. The Godel sentence, like the parallel postulate, is true in the standard model and false in non-standard ones. For anybody to call G "true" is even more outrageous than usual, because at the relevant cardinality (aleph-0), there is, up to isomorphism, ONLY ONE model in which G is true -- it is false in ALL the others! Saying of any sentence that is "demonstrably" true, in SOME system, without first imposing relevant constraints on the kind of system, is even more idiotic -- obviously every sentence s is "demonstrably true" in the system S whose only axiom is s. > In his book on GIT, the late TF (to his great credit) challenges > [quoted text clipped - 6 lines] > This forces us to ask, "Does Prof.Peter Smith know Prof.Daniel > Isaacson?" Oh? Why does it force us to ask that?? Actually I do know Dan, and of course we'd both agree with Torkel Franzen's point. (It is one anyone teaching this stuff stresses: sure, we need to assume that the system we are dealing with is indeed consistent if we are get to see that the canonical Gödel sentence is true on the standard interpretation.) > Because Isaacson, TEACHING THIS STUFF RIGHT NOW ("Michaelmas > term, 2007), flaunts his ignorance of this Torkelian perspective, > lecturing, > >>>>> The Godel sentence is demonstrably true, > >>>>> though not demonstrable in the system for which it is constructed. Hold on, hold on. You've rather ripped this out of context from Dan's lecture notes. He two paragraphs earlier introduces the assumption that we are dealing with systems of arithmetic S which are *sound* (and hence consistent). And on *that* assumption, which I take to be still in force when we comments on the proof he sketches, the canonical Gödel sentence will indeed be true on the standard interpretation. > > In his book on GIT, the late TF (to his great credit) challenges > [quoted text clipped - 26 lines] > canonical Gödel sentence will indeed be true on the standard > interpretation. It does not matter that the system is "sound" i.e. it does not help to introduce yet another concept. If we can formally prove S to be consistent we can only do it in another system, whose consistency we not know. The "proof" is an empty game with symbols. It does NOT tell that no matter how long we keep generating theorems in S we will never derive a contradiction. It has zero cogency. If we are 100% sure that a system is consistent then it cannot be because of a formal proof. There are three possibilities: 1) We do not have the foggiest idea if PA is consistent and we will never know. Hence we do not know if Gödel sentence is true. 2) The human mind surpasses any computer 3) There exists a formalization of arithmetic that can prove its own consistency. But it is not possible to reject 1 and 3 and at the same time claim that the human mind does NOT surpass any machine. > There are three possibilities: > 1) We do not have the foggiest idea if PA is consistent and we will [quoted text clipped - 5 lines] > But it is not possible to reject 1 and 3 and at the same time claim > that the human mind does NOT surpass any machine. In the strictest sense, the correct answer is 1). But: if PA should ever turn out to be inconsistent, then the whole subject of foundations of mathematics as we now know it will have to be redone: we assume everywhere in mathematics that the natural number sequence makes sense, and that it has the PA properties. We even presume it in definitions of the formal language of FOL. In short: 1) is correct: we only -do as if- we know PA is consistent, and we do so for pragmatic reasons. SignatureCheers, Herman Jurjus >> There are three possibilities: >> 1) We do not have the foggiest idea if PA is consistent and we will [quoted text clipped - 15 lines] > In short: 1) is correct: we only -do as if- we know PA is consistent, > and we do so for pragmatic reasons. I totally agree with you. If PA is inconsistent, it might be so without us knowing why: the inconsistency proof might be quite literally beyond human grasp! (One of the fallacy easy to fall into is we somehow know *all* the finite information!). For what's it worth, if PA is inconsistent, 2+2=4 is still "true": it's just PA wouldn't have any model to contain such truth, simply because it doesn't have any model at all. > I totally agree with you. If PA is inconsistent, it might be so without > us knowing why: the inconsistency proof might be quite literally beyond > human grasp! (One of the fallacy easy to fall into is we somehow know > *all* the finite information!). That is NOT a fallacy. Just because we don't know it YET does NOT mean we CANnot know it. Just because it is beyond our CURRENT grasp does NOT mean it is beyond our REACH! Your invovling humans in any of this at all is just more evidence of your complete unfitness to even be participating. On Dec 8, 3:41 am, "Nam D. Nguyen" <namducngu...@shaw.ca> wrote: > I totally agree with you. If PA is inconsistent, it might be so without > us knowing why: the inconsistency proof might be quite literally beyond > human grasp! (One of the fallacy easy to fall into is we somehow know > *all* the finite information!). The assertion that PA is inconsistent is equivalent to the assertion that a certain diophantine equation has a solution. If PA is inconsistent, there is a solution and it cannot be in principle beyond human grasp as all it involves doing is brute calculation. I also thought I should point out that if one finds manifestly true axioms T which prove con(PA), and carries out a proof using those axioms, then con(PA) will be a settled matter, in the same way that theorems such as Fermat's Little Theorem, and the law of Quadratic Reciprocity are settled matters. There is no basis for saying "but what about con(T)?" That may be a theorem which one might try to prove in the manner one proved con(PA), but it is of no relevance to the truth of con(PA) which follows from the manifest truth of the axioms which suffice to prove it. The fact that there is a hypothetical manner in which con(PA) might become a settled matter is often underemphasised. On Dec 8, 11:52 am, kleptomaniac6...@hotmail.com wrote: > I also thought I should point out that if one finds manifestly true > axioms T Well, there's no need to worry about THAT. There are no manifestly true axioms. EVERY axiom-set that is not inconsistent is manifestly true in SOME model. Its denial therefore is as well. They obviously CAN'T BOTH be "manifestly" true. > On Dec 8, 11:52 am, kleptomaniac6...@hotmail.com wrote: > [quoted text clipped - 7 lines] > Its denial therefore is as well. > They obviously CAN'T BOTH be "manifestly" true. Okay, if I wrote that post again I would use not the phrase "manifestly true axioms" but rather the phrase "axioms whose intended meaning is manifestly true". I hope the rest of my post didn't make you too angry. > On Dec 8, 3:41 am, "Nam D. Nguyen" <namducngu...@shaw.ca> wrote: > > I totally agree with you. If PA is inconsistent, it might be so [quoted text clipped - 7 lines] > The assertion that PA is inconsistent is equivalent to the assertion > that a certain diophantine equation has a solution. Agree. (I don't think I've contradicted this, or any equivalent fact). > If PA is > inconsistent, there is a solution and it cannot be in principle beyond > human grasp as all it involves doing is brute calculation. The point of contention here is what exactly we do mean by: "it cannot be in principle beyond human grasp" I could easily come up with an Principle to the effect that there is a mathematical truth that: "can be in principle beyond human grasp"! I think I did state that principle a couple of times in the past. > The point of contention here is what exactly we do mean by: > [quoted text clipped - 6 lines] > > I think I did state that principle a couple of times in the past. To your unending embarrassment. On Dec 8, 11:41 am, kleptomaniac6...@hotmail.com wrote: > If PA is > inconsistent, there is a solution and it cannot be in principle beyond > human grasp as all it involves doing is brute calculation. The important part there is "in principle". It is not clear what the relevant principles ought to be because it is not clear how great the species is going to grow. Theories-satisfying-the-hypotheses-of-G1, i.e., theories that are recursively axiomatizable but r.e as OPPOSED to (totally) recursive, MUST have SOME short theorems with VERY long proofs. NN's point is that the AMOUNT of calculation required might be, DESPITE being finite, beyond human achievement. In real life, this question is not even worthy of being speculated upon. NN is baically an idiot for even caring. Or rather, for continuing, after a decade, to still be mistaking this ("finite but humanly impossible") for a possible degree of difficulty. We CARE about what TMs can finitely achieve. WE DON'T CARE about what humanity may eventually achieve. > On Dec 8, 11:41 am, kleptomaniac6...@hotmail.com wrote: >> If PA is [quoted text clipped - 8 lines] > (totally) recursive, MUST have SOME short theorems with > VERY long proofs. > NN's point is that the AMOUNT of calculation required might be, > DESPITE being finite, beyond human achievement. That's my point. Though it's only half of what I'm trying to say. (And I'll come to the other half shortly). > In real life, this question is not even worthy of being speculated > upon. In "real life", we haven't solved the GC question. So, should ~GC be provable in Q, the number of digits of a prime involved in the shortest proof could conceivably be literally beyond human grasp! Which means the entire human mathematical reasoning effort would fail to know the theorem-hood of an F whose form and semantics is arguably slightly more complex than "2+2=4"! That alone would be worthy of being mentioned, I'd imagine. > NN is baically an idiot for even caring. Not sure on that; but I do think one should care to listen to the other half of the GC story, which points to a even more somber note on the limit of FOL's finite-reasoning. If ~GC is genuinely not provable in Q, it's *impossible* to know the decidability of GC in Q, via FOL's finite-reasoning! > Or rather, for continuing, after a decade, to still be mistaking > this ("finite but humanly impossible") for a possible degree of > difficulty. Whatever that might mean. > We CARE about what TMs can finitely achieve. WE DON'T CARE > about what humanity may eventually achieve. Don't forget TM is an abstract product, tool of *human* mind! On Dec 8, 5:41 pm, kleptomaniac6...@hotmail.com wrote: > On Dec 8, 3:41 am, "Nam D. Nguyen" <namducngu...@shaw.ca> wrote: > > I totally agree with you. If PA is inconsistent, it might be so [quoted text clipped - 9 lines] > inconsistent, there is a solution and it cannot be in principle beyond > human grasp as all it involves doing is brute calculation. Well, you have to find the solution, first, before doing the brute calculation. So I'm not sure what diophantine equations bring to your argument. If PA is inconsistent, there exists a proof of "not 0 = 0". Once you find the proof, it is just a matter of checking to see that it is a solution. >> > >>>>> The Godel sentence is demonstrably true, >> > >>>>> though not demonstrable in the system for which it is constructed. [...] >It does not matter that the system is "sound" i.e. it does not help to >introduce yet another concept. If we can formally prove S to be [quoted text clipped - 3 lines] >derive a contradiction. It has zero cogency. If we are 100% sure that >a system is consistent then it cannot be because of a formal proof. You've changed the subject by introducing the concept of "being 100% sure." The original statement under discussion said only that the Goedel sentence is demonstrably true, which it is: There does indeed exist a demonstration of its truth. What attitude we take towards that demonstration is a separate question. You can choose to doubt it, but the "it" that you're doubting still exists (otherwise, what is it that you're doubting?). SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences "Newberry" <newberryxy@gmail.com> ha scritto >There are three possibilities: >1) We do not have the foggiest idea if PA is consistent and we will >never know. Hence we do not know if Gödel sentence is true. >2) The human mind surpasses any computer >3) There exists a formalization of arithmetic that can prove its own >consistency. You could think that the human mind surpasses PA (in the sense that it can convince itself of thing not provable in PA) but (for example) does not surpass other unknown extensions of PA. For example there could be a *true* arithmetical statement U about which human mind cannot ever be convinced. So the human mind would never surpass PA+U, and will also never know which statement is U. Newberry says... >There are three possibilities: >1) We do not have the foggiest idea if PA is consistent and we will >never know. Hence we do not know if the Goedel sentence is true. >2) The human mind surpasses any computer >3) There exists a formalization of arithmetic that can prove its own >consistency. I believe that all three of those are false, so I reject your claim that those are the only three possibilities. Your number 1 assumes that: To know X means that X is provable in some system T0 such that T0 is provably consistent in some system T1 such that T1 is provably consistent in some system T2 such that ... By that definition, we never know anything. I suppose that might be correct, in some sense, but the way most people use the word "know" is something short of proof. Or I should say *different* from proof. Proof is neither necessary nor sufficient for knowledge. In any case, whether or not we can be said to *know* with certainty that PA is consistent, it is certainly false to say that "we do not have the foggiest idea if PA is consistent". That's completely wrong. The consistency of PA is as certain as *anything*. There is no reason to believe otherwise, and plenty of reason to believe it. >But it is not possible to reject 1 and 3 and at the same time claim >that the human mind does NOT surpass any machine. Sure, it's possible. I reject 1 and 3 and at the same time calim that the human mind does not surpass any machine. -- Daryl McCullough Ithaca, NY > Newberry says... > [quoted text clipped - 36 lines] > Daryl McCullough > Ithaca, NY That seems a pretty good diagnosis. It does seem, indeed, that some of the suspect claims people make e.g. about our supposedly not be able know that PA is consistent have nothing much to do the philosophy of mathematics itself, but rather arise from importing the sort of dubious assumptions about what is required for knowledge generally that leads to rampant across-the-board scepticism. > > Newberry says... > [quoted text clipped - 43 lines] > dubious assumptions about what is required for knowledge generally > that leads to rampant across-the-board scepticism. Sigh. > It does seem, indeed, that some of > the suspect claims people make e.g. about our supposedly not be able > know that PA is consistent have nothing much to do the philosophy of > mathematics itself, Society advances 1 funeral at a time. Doubts about the consistency of systems satisfying the hypotheses of G1 come from 1 very well-known place: infinity. That is especially relevant since proofs HAVE to be finite. G1 is part of a cluster of results implying, basically, that a pre- requisite for the existence of a finitary description of an infinite thing is that the infinite thing be recursive. If it is merely (like 1st-order theories) rec.enum. instead, then there is definitionally always doubt. In BOTH directions. > In any case, whether or not we can be said to *know* with > certainty that PA is consistent, it is certainly false to > say that "we do not have the foggiest idea if PA is consistent". > That's completely wrong. The consistency of PA is as certain > as *anything*. Well, it seems to me that this last assertion is completely wrong. "2 + 2 = 4" seems more certain to me than the consistency of PA, and I imagine the same is true for you, as well. After all, the consistency of PA - that a particular logico-mathematical system does not ever produce among its deductions "not 0 = 0" - presumably depends on an argument for you to believe it. "2 + 2 = 4" does not. > There is no reason to believe otherwise, and > plenty of reason to believe it. There is plenty of reason to believe that water boils at 100 degrees centigrade, and no reason to believe otherwise. That hardly means that it is "as certain as anything." pboparis@gmail.com says... >> In any case, whether or not we can be said to *know* with >> certainty that PA is consistent, it is certainly false to [quoted text clipped - 5 lines] >+ 2 = 4" seems more certain to me than the consistency of PA, and I >imagine the same is true for you, as well. Well, I should probably say as certain as any nontrivial mathematics. I don't think it is possible to do anything nontrivial without PA (or something equivalent). >After all, the consistency of PA - that a particular >logico-mathematical system does not ever produce among >its deductions "not 0 = 0" - presumably depends on an >argument for you to believe it. "2 + 2 = 4" does not. Sure it does. >> There is no reason to believe otherwise, and >> plenty of reason to believe it. > >There is plenty of reason to believe that water boils at 100 degrees >centigrade, and no reason to believe otherwise. That hardly means >that it is "as certain as anything." Well, I'm mainly taking issue with the claim that "we do not have the foggiest idea if PA is consistent". That's not true. -- Daryl McCullough Ithaca, NY > Well, I'm mainly taking issue with the claim that "we do not > have the foggiest idea if PA is consistent". That's not true. Who is claiming that? TF? I never saw him entertain that seriously in life. The book just says that "in general", right? Not for PA specifically, right? In any case there do exist consistency proofs for PA in systems of set theory that do seem more than just "foggily" sound. So we do have an idea. It's just not the kind of idea that deserves any sort of mathematical respect. My personal opinion is that mathematical respect flows out of the completeness theorem: IF it is inconsistent, THEN THAT MUST be provable. Therefore, the BURDEN of proof rests ALWAYS UPON people expressing doubts about consistency. If their doubts are at all justified then a proof must in fact exist, so they are obligated to bring it back alive -- that demand is reasonable if their position is justifiable. SO UNTIL then, THEY should STFU. >> Well, I'm mainly taking issue with the claim that "we do not >> have the foggiest idea if PA is consistent". That's not true. [quoted text clipped - 13 lines] > completeness theorem: IF it is inconsistent, THEN THAT MUST be > provable. Why "must", a _subjective_ verb? What happens if such inconsistency proof is beyond human reach? > Therefore, the BURDEN of proof rests ALWAYS UPON people expressing > doubts about consistency. But what happens if the theory is genuinely consistent but it's *impossible* to know that? And if it's impossible to know the consistency then isn't it true we'd not know the state of inconsistency, and therefore might harbor the doubt that it might be inconsistent, logically speaking? > If their doubts are at all justified then a proof must in fact exist, > so they are obligated to bring it back alive -- that demand is > reasonable if their position is justifiable. "Doubt" doesn't have to follow dualism-rule: I could doubt *both* opposite claims simply because I might not know enough about any of them. For instance, I doubt if ~GC is provable in Q and I also doubt GC is provable in Q! (Odd huh? But could one do better than I've done?) > > My personal opinion is that mathematical respect flows out of the > > completeness theorem: IF it is inconsistent, THEN THAT MUST be > > provable. > > Why "must", a _subjective_ verb? That is NOT a subjective verb. That is an OBJECTIVE must. It really must. Just as surely as 2+2 MUST be 4 and not 3 or 5. > What happens if such inconsistency proof > is beyond human reach? You have no way of knowing that ANYthing is beyond human reach. You don't know how many humans there are going to eventually be. You don't know how big human brains are eventually going to get. It is at least theoretically possible that there is no upper limit on the number of humans, that the human population will keep increasing exponentially throughout all currently known galaxies and that just about the time it appears we are about to fill them all up, some sort of cosmic revelation will occur and more space will become available. Or a similar sort of revelation will cause there to be an infinite number of humans. Or cause some individual finite-brained human parents to have human children with an infinitary brain-part. "beyond human reach" IS JUST STUPID. The issue in any case IS NOT what "humanity" can do BUT RATHER what A TURING MACHINE can do. > > Therefore, the BURDEN of proof rests ALWAYS UPON people expressing > > doubts about consistency. > > But what happens if the theory is genuinely consistent but it's *impossible* > to know that? It by definition CANNOT be impossible. If a recursive set of first-order sentences is inconsistent then THERE IS A FINITE proof of that. That is a FACT. That is a THEOREM (the completeness theorem). Sorry your ignorant a.s was still too ignorant to know that after all these years. You just plain should've QUIT a long time ago. You recently promised to. Please hold yourself to it. >>> My personal opinion is that mathematical respect flows out of the >>> completeness theorem: IF it is inconsistent, THEN THAT MUST be >>> provable. >> Why "must", a _subjective_ verb? > > That is NOT a subjective verb. That is an OBJECTIVE must. There is nothing as an "objective must" in mathematical reasoning. Given a T and an F in L(T), F is or is not a theorem of T. No "must" is *required* here, of course! > It really must. Just as surely as 2+2 MUST be 4 and not 3 or 5. One could *easily* come up with a formal system in which 2+2 is 3. So your "as sure as" is not a mathematical certainty. >> What happens if such inconsistency proof >> is beyond human reach? > > You have no way of knowing that ANYthing is beyond human reach. On the contrary, We always know *something* (like the *existence of something*) that is beyond human reach - even human with *an* infinite-knowledge! I give you 2 hints: - Think of the phrases such as "cardinality" and "Power Set Axiom" - Any reasoning framework must necessarily be based on knowledge of certain cardinality. (In the case of FOL, for example, proofs are based on *finite* cardinalities.) > You don't know how many humans there are going to > eventually be. You don't know how big human brains are eventually [quoted text clipped - 7 lines] > infinite > number of humans. > Or cause some individual finite-brained human parents > to have human children with an infinitary brain-part. "human ... with an infinitary brain-part" is at best imaginative (i.e. not real) and at worst is ignorant of what we currently know about our Universe: in a finite number years, it either would vanish in a Big Crunch, or would settle down in a state in mode in which the only thing left is not matter but quanta at lowest possible states, and in which case no "human" could possibly exist! > "beyond human reach" IS JUST STUPID. Think twice before capitalizing any word! What word one capitalizes might de-capitalize one's knowledge on the backfire! > The issue in any case IS NOT what "humanity" can do BUT RATHER what > A TURING MACHINE can do. In mathematical reasoning context, "what A TURING MACHINE can do" is just another way of saying "there exists ...". But since mathematical reasoning is meaningless without human beings [think about the word "reasoning"! Who would be the reasoner(s)?], brining the catch-phrase "TURING MACHINE" doesn't make this case less about "human"! >>> Therefore, the BURDEN of proof rests ALWAYS UPON people expressing >>> doubts about consistency. [quoted text clipped - 5 lines] > IS A FINITE proof of that. That is a FACT. That is a THEOREM (the completeness > theorem). Why is it that I talked about _consistency_ and you about _inconsistency_? > Sorry your ignorant a.s was still too ignorant to know that after all > these years. Seems you've not changed in all these years: vulgar in communication, impulsive in quick reaction to see what the opponents might be *really* talking about! > You just plain should've QUIT a long time ago. You recently promised > to. Please hold yourself to it. I do quit sharing the plan of discussing with this forum on the "revamp" of FOL, mainly because people don't seem to care much about any change to FOL. But this is Holiday Season and I got some spare time so I don't think it's much of a "violation" of my pledge, to occasionally jump in when I think the poster is too wrong on something I think important. (Of course I stand to be corrected where I'm technically wrong). > There is nothing as an "objective must" in mathematical reasoning. Of course there is. That is BY DEFINITION what mathematical reasoning is ABOUT. The fact that you didn't know this IS WHY you should sit down and shut up. > Given a T and an F in L(T), F is or is not a theorem of T. No "must" > is *required* here, of course! OF COURSE it is. IF F is a theorem of T, then F *must* be a theorem of T. That is NOT contingent. It IS NOT like it COULD have turned out NOT to be a theorem, like there is some alternative universe in which it is NOT a theorem. IF p==>q then p MUST entail q. INside the system, logical consequence IS MODAL. It MEANS "must". > >> But what happens if the theory is genuinely consistent but it's *impossible* > >> to know that? I mistakenly replied, > > It by definition CANNOT be impossible. > > If a recursive set of first-order sentences is inconsistent then THERE > > IS A FINITE proof of that. That is a FACT. That is a THEOREM (the completeness > > theorem). > > Why is it that I talked about _consistency_ and you about _inconsistency_? Because I was giving you credit for having half a brain. Nobody (except you) is ever stupid enough to ask, "But whappens if the sky is blue?". I assumed you were asking something that people might actually question. That "the theory is consistent but it's impossible to know that" IS PRECISELY THE SITUATION WE HAVE ALL OBVIOUSLY BEEN IN ever since 1931! Nobody EVER asks what happens *if* that is the case! That FACTUALLY IS KNOWN to be the case! What IS HAPPENING is what happens in THAT case! > Seems you've not changed in all these years: vulgar in communication, impulsive > in quick reaction to see what the opponents might be *really* talking about! You are not really talking about anything. You never have. You are interested in a class of question that is simply not important. > >> Well, I'm mainly taking issue with the claim that "we do not > >> have the foggiest idea if PA is consistent". That's not true. [quoted text clipped - 16 lines] > Why "must", a _subjective_ verb? What happens if such > inconsistency proof is beyond human reach? Why then, as far as we humans are concerned, the system is consistent after all. So there's nothing to worry about. > > Therefore, the BURDEN of proof rests ALWAYS UPON people > > expressing doubts about consistency. > > But what happens if the theory is genuinely consistent but it's > *impossible* to know that? Under George's philosophy, we assume it consistent by default, since we have found no proof to the contrary. On Dec 8, 9:28 pm, be...@pop.networkusa.net wrote: > > >> Well, I'm mainly taking issue with the claim that "we do not > > >> have the foggiest idea if PA is consistent". That's not true. [quoted text clipped - 28 lines] > Under George's philosophy, we assume it consistent by default, since > we have found no proof to the contrary. What if an argument that the assumption that there is truth implies there is a predicate "true", and then in quantifying over objects, they all satisfy it so there is unrestricted comprehension. Then, in building sets, if there is some supertheory of ZFC containing ZFC's axioms as thus truisms, with the unrestricted comprehension over sets in ZFC, then there's a universal collection, a collection of all the elements of ZFC. Yet, then Cantor/Russell etcetera as paradoxes follow. Otherwise there's not a universal quantifier (with considerations of around three different kinds of "universal" quantifiers, indicating various accords or lack thereof with the transfer principle, for any / for each / for every / for all.) Among reasons I think ZF is inconsistent, consider any theory that has as its elements of discourse those elements of ZF (casually referring to ZF as a collection of sets defined by the set-theoretical non- logical/proper axioms), and as well some other elements. Now, quantifying over those elements with "x is a set in ZF", then that collection is the Russell set, so the set containing "x: x is a set in ZF" is the Russell set so ZF contains an irregular set. Otherwise there's no universe (in a broad sense). People seem quick to accept that the Russell set contains unspecified elements but few address the objects of Peano Arithmetic notionally having a similar concern. Consider Burali-Forti, that the order type of ordinals would be an ordinal so there is no collection of all ordinals in ZF, in terms of a difference between "for any", "for each", "for every", and for "all". For each ordinal, its order type is an ordinal, for all ordinals, their order type is not a set. That's basically in distinction of the transfer principle and making shorthand the notion of arbitrarily extended induction, particularly those structures that are only primitively distinguishable among themselves via induction, a memoryless two-step process. It seems those natural objects bootstrap themselves (hoist by their own bootstraps) into a framework where they have a synthetic interface. That is to say, the natural integers form naturally, as a consequence there is infinity, and the universe, and only after where there is the complete framework of all objects is it possible to synthetically distinguish two integers. They do so from nothing. ZF has no universe. In that sense ZF isn't, for example, a Cantorian set theory, where Cantor wanted both a universe and infinite powerset incongruence in his theory. Those two were found incompatible, thus the universe was discarded. There definitely, by definition, specifically, is a universe where the domain of discourse is no other thing. No theory exists in a vacuum. Then, in consideration of which axioms of ZF might be false, I think the axiom of infinity is incorrectly stated, because a variety of fundamental theorems of a set-theoretical infinity, among all possible set theories, would have that infinity is non-well-founded, and in large structures their grandness presupposes their identity. Then, that would lead to the notion that ZF's axiom of regularity is as well so not-necessarily-true: false. There are no universal truths in a theory without the universe, and where there's a universe it's THE universe. Ross -- Finlayson Consulting > On Dec 8, 9:28 pm, be...@pop.networkusa.net wrote: > [quoted text clipped - 88 lines] > There are no universal truths in a theory without the universe, and > where there's a universe it's THE universe. Well, if you just add elements a, b, c beside the empty set will it make the theory inconsistent? > Ross > > -- > Finlayson Consulting- Hide quoted text - > > - Show quoted text - > > What if an argument that the assumption that there is truth implies > > there is a predicate "true", and then in quantifying over objects, [quoted text clipped - 7 lines] > > quantifiers, indicating various accords or lack thereof with the > > transfer principle, for any / for each / for every / for all.) ... > > People seem quick to accept that the Russell set contains unspecified > > elements but few address the objects of Peano Arithmetic notionally [quoted text clipped - 14 lines] > > there is the complete framework of all objects is it possible to > > synthetically distinguish two integers. They do so from nothing. ... > Well, if you just add elements a, b, c beside the empty set will it > make the theory inconsistent? I'm not quite sure. If there are simply three constants labelled a, b, c, besides the empty set, does that presume a set theory? Are a, b, and c sets or primitives/ur-elements? I see powerset and union as operations available in any set theory, not requiring axiomatization, because any cognizance of the plurality of elements of the theory leads to those constructs via comprehension, which must exist else the constants would be inaccessible to formulae, thus anonymous and sans label, indistinguishable. The existence of composites is implicit. Basically it seems that the theory is "c =/= a = a =/= b = b =/= c =", with there only being identity, tautology. From thinking about the foundations of mathematical logic, reduction via deduction led me to think that there is reason to consider that just starting from nothing, in set theory the empty set, that there is a natural tendency towards diversity, maximality, and a dichotomous "force" towards simplicity, minimality. Then, from the "what if?" question applied to the empty set, there is something else, and then something neither of those etcetera ad infinitum, a continuum of primitive and ur-elements, with there being only one variable/constant _the_ ur-element or proto-element. That's just the way it is. Then, that theory exists because it is basically totally unrestricted comprehension, and from its fabric, any other theory's elements would find a natural image, consistent theories in completion and inconsistent theories, well, incompletion, having referents to extra- theoretical elements inaccessible from the theory. So, where the theory is "there exist three distinct things: a, b, and c", it seems true already. Yet, there exist(s) four distinct things, five distinct things, etcetera. In a way that's the notion that the integers exist, each integer exists. Ross -- Finlayson Consulting > > > What if an argument that the assumption that there is truth implies > > > there is a predicate "true", and then in quantifying over objects, [quoted text clipped - 63 lines] > five distinct things, etcetera. In a way that's the notion that the > integers exist, each integer exists. OK, let's say a, b, c, aa, ab, ac, ba, bb, ba, ca, cb, cc, aaa ... > > > > What if an argument that the assumption that there is truth implies > > > > there is a predicate "true", and then in quantifying over objects, [quoted text clipped - 65 lines] > > OK, let's say a, b, c, aa, ab, ac, ba, bb, ba, ca, cb, cc, aaa ... Are they all the words in the alphabet {a, b, c}, i.e. Sigma* for Sigma = {a, b, c} (including the empty set)? Then, I would consider whether there are then meaningful descriptions of the words, such as their lengths, that can be inferred to hold in the manner of distinguishing the words from each other. It seems you describe a theory with an infinite list of constants, well represented as a trinary rooted tree with the root being the empty set and each node having three leaves with each edge as operation concatenating the value of the parent node with one of a, b, and c, with each label applying to only one child edge of a node, in a compact description. Then, does the theory of an infinite collection of constants, described in that manner as closure of the empty set with regards to the operation of appending one of a, b, and c, bring in all manner of combinatoric results as baggage of the description? Those constants aren't just distinguishable, they are partitionable, by any of a variety of methods, where their label is a composite, with its constituent elements distinguishable. As to whether any possible initial segment of an enumeration of constants is partitionable, that begs the definition of partitionable. I wonder about some collection of constants such that each was distinguishable yet no predicate partitions them in any manner besides as singletons. That leads to various reflections on the dually-self-infraconsistent. Ross -- Finlayson Consulting > On Dec 8, 9:28 pm, be...@pop.networkusa.net wrote: > [quoted text clipped - 39 lines] > of all the elements of ZFC. Yet, then Cantor/Russell etcetera > as paradoxes follow. No; that would require unlimited comprehension over the sets of the new theory as well. [...] > Among reasons I think ZF is inconsistent, consider any theory > that has as its elements of discourse those elements of ZF [quoted text clipped - 4 lines] > the set containing "x: x is a set in ZF" is the Russell set so > ZF contains an irregular set. No, you can't do that in ZF; you can only get "x e S: x is a set in ZF" for any set S in ZF. This is just S itself of course. > Otherwise there's no universe (in a broad sense). There's no universe in ZF; this is easily proven by the Russell argument. > People seem quick to accept that the Russell set contains > unspecified elements but few address the objects of Peano [quoted text clipped - 4 lines] > for "all". For each ordinal, its order type is an ordinal, for > all ordinals, their order type is not a set. That isn't standard usage of "for all" in mathematics, I'm afraid. You'd need to say something like "the order type of the collection of all ordinals is not a set". I snipped the rest of your post; but it isn't a demonstration that ZF is inconsistent, only an argument that its axioms aren't obviously true. If I accepted your argument I would be less convinced in ZF's consistency; but there's still George's point that noone has ever found an inconsistency. This is an empirical claim and does not require philosophical justification. On Dec 7, 10:44 pm, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > pbopa...@gmail.com says... Who was me - one of my sons had changed my login on my computer. Grrr... > >> In any case, whether or not we can be said to *know* with > >> certainty that PA is consistent, it is certainly false to [quoted text clipped - 8 lines] > Well, I should probably say as certain as any nontrivial > mathematics. That's better. On the other hand, there's still a difference between, "There's a prime number between 10 and 20" and "For every natural number n there's a prime number greater than n." > I don't think it is possible to do anything > nontrivial without PA (or something equivalent). Do you consider Quadratic Reciprocity or Bertrand's Postulate non- trivial? Both can be proved without using PA, that is in second-order PA \ {successor axiom}, which has as a model all initial segments as well as the standard model. I suspect Fermat's Last Theorem (which is surely non-trivial?) can be proven in this reduced theory as well. > >After all, the consistency of PA - that a particular > >logico-mathematical system does not ever produce among > >its deductions "not 0 = 0" - presumably depends on an > >argument for you to believe it. "2 + 2 = 4" does not. > > Sure it does. Really? You only believe "2 + 2 = 4" because of an argument? That is, I'm not denying that you *can* use an argument (a Principia Mathematica-style proof) to arrive at "2 + 2 = 4". I'm saying that's not the actual, causal reason why you believe it. Of course, you might be different from me in this matter, but I would be surprised if you are (and quite interested if you are!). >Well, I should probably say as certain as any nontrivial >mathematics. I don't think it is possible to do anything >nontrivial without PA (or something equivalent). What I am about to say is a quibble, but perhaps an interesting one. Work in reverse mathematics, as exemplified in Steve Simpson's book "Subsystems of Second-Order Arithmetic," has shown that indeed one can do quite a lot in systems weaker than PA. Five systems are studied in that book, the three weakest being RCA_0, WKL_0, and ACA_0, in increasing order of strength. ACA_0 is essentially equivalent to PA. A substantial amount of mathematics can be built up on the basis of RCA_0. Think of a theorem that does not obviously depend essentially on uncountable sets or countable choice, and there is a good chance that it is a theorem of RCA_0. To get things like Goedel's completeness theorem or Brouwer's fixed-point theorem you need to ascend to WKL_0, but you don't need ACA_0. More generally, PA gives you induction for *all* first-order formulas, and for many applications that's a lot stronger than necessary. It is possible to weaken the induction axiom even further to get systems weaker than RCA_0 (such as the system that Harvey Friedman calls exponential function arithmetic, a.k.a. IDelta_0(exp)) and still recover a lot of nontrivial mathematics. SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences On Dec 8, 5:51 pm, tc...@lsa.umich.edu wrote: > In article <fjcerj01...@drn.newsguy.com>, > [quoted text clipped - 14 lines] > To get things like Goedel's completeness theorem or Brouwer's fixed-point > theorem you need to ascend to WKL_0, but you don't need ACA_0. I think the point you are making against Daryl is reasonably vacuous Consider any arithmetical theorem which is non-trivial and which can be proved in first-order PA. Then the number of appeals to the induction axiom is finite, and so voila by considering PA \ {all instances of induction} + {instances of induction needed to prove the theorem}, one arrives at a sub-theory of PA which can prove a non- trivial theorem. The interesting question (in my eyes anyway) is what one can prove without having to assume that the numbers are "all" there, which PA, as well as all these other second-order systems you've mentioned, assumed. > The interesting question (in my eyes anyway) is what one can prove > without having to assume that the numbers are "all" there, which PA, > as well as all these other second-order systems you've mentioned, > assumed. All where? On Dec 7, 2:07 pm, pbopa...@gmail.com wrote: > Well, it seems to me that this last assertion is completely wrong. "2 > + 2 = 4" seems more certain to me than the consistency of PA, and I > imagine the same is true for you, as well. Well, 2+2=4 is a theorem of PA. Con(PA) is not. So of course it seems more ceratin. > After all, the consistency > of PA - that a particular logico-mathematical system does not ever > produce among its deductions "not 0 = 0" - presumably depends on an > argument for you to believe it. "2 + 2 = 4" does not. Yes, actually, it does. That theorem has a proof just like any other. The fact that you went through the argument in 1st grade does not end the existence of the argument. > There is plenty of reason to believe that water boils at 100 degrees > centigrade, and no reason to believe otherwise. That hardly means > that it is "as certain as anything." There is plenty of reason not to bring the physical world into this. > Newberry says... > [quoted text clipped - 32 lines] > Sure, it's possible. I reject 1 and 3 and at the same time calim > that the human mind does not surpass any machine. How can that possibly be? Even TF says that PA is consistent, ZFC is consistent. PA consistency can be proved in ZFC (or a fragment thereof), but the consistency of ZFC can be proved only by using axioms of infinity, which are dubious. So we cannot prove the consistency of ZFC. Yet we are sure of its consistency. So how come we are sure if we cannot prove it?? Contrary to your earlier claim, the following proof is not formalizable: 1. The axioms of PA (ZFC) are manifestly true 2. Truth cannot be inconsistent with itself 3. Hence PA (ZFC) is consistent for if it were provable in some system S, S would be incomplete and its consistency unknown, hence the cogency of the proof would converge to zero. (I think you admitted this much yourself.) The system is sound? Is this argument supposed to appeal to intuition? Fine. But once you formalize it in a system S ... Newberry says... >How can that possibly be? Even TF says that PA is consistent, ZFC is >consistent. PA consistency can be proved in ZFC (or a fragment >thereof), but the consistency of ZFC can be proved only by using >axioms of infinity, which are dubious. So we cannot prove the >consistency of ZFC. Yet we are sure of its consistency. So how come we >are sure if we cannot prove it?? That's a pychological question, not a mathematical question. It just shows that certainty does not always come from proof. We are convinced about the truth of Peano Arithmetic because we have experience with arithmetic, and all the axioms of PA seem to be true, according to our experienc >Contrary to your earlier claim, the following proof is not >formalizable: >1. The axioms of PA (ZFC) are manifestly true >2. Truth cannot be inconsistent with itself >3. Hence PA (ZFC) is consistent Well, you're wrong. It's perfectly formalizable. It's been *formalized*. >for if it were provable in some system S, It is. >S would be incomplete and its consistency unknown, Yes, S would be incomplete, but no, its consistency would not be unknown. What you can conclude, based on Godel's theorem, plus some plausible assumptions about what human mathematicians find convincing, is that *if* we are convinced that a theory S is true (or at least, that what it says about arithmetic is true), then we are also convinced about the truth of statements that are not provable by S. From this it follows that if a recursively enumerable theory S could prove *all* the arithmetical statements that a human mathematician would ever find convincingly true, then we would not be able to become absolutely convinced of the consistency of S. -- Daryl McCullough Ithaca, NY On Dec 8, 12:43 pm, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 10 lines] > we have experience with arithmetic, and all the axioms of > PA seem to be true, according to our experienc So we are convinced about the truth of Peano Arithmetic not because of a formal proof i.e. we surpass any machine. > >Contrary to your earlier claim, the following proof is not > >formalizable: [quoted text clipped - 30 lines] > Daryl McCullough > Ithaca, NY Newberry says... >On Dec 8, 12:43 pm, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> That's a pychological question, not a mathematical question. >> It just shows that certainty does not always come from proof. [quoted text clipped - 4 lines] >So we are convinced about the truth of Peano Arithmetic not because of >a formal proof i.e. we surpass any machine. Why do you say that? We're convinced for psychological reasons, not proof. How does that show that we surpass any machine? That doesn't make a bit of sense. -- Daryl McCullough Ithaca, NY > Newberry says... > [quoted text clipped - 12 lines] > not proof. How does that show that we surpass any machine? That > doesn't make a bit of sense. I think you would agree that the consistency of PA is a hard fact not just an emotional state or a similar psychological phenomenon. It means that no matter how long and in what order we keep generating theorems we will never derive P & ~P. PA can be literally materialized as a mechanical system, a machine. We are asking if tis machine can ever produce P & ~P. No machine can answer that question. We are able to answer it, hence it we surpass any machine. I do not see what the introduction of the term "psychological" has to do with this. > -- > Daryl McCullough > Ithaca, NY Newberry says... >I think you would agree that the consistency of PA is a hard fact not >just an emotional state or a similar psychological phenomenon. The consistency of PA is a hard fact. The fact that I *consider* it to be a hard fact is a psychological phenomenon. It doesn't in any way show that there is something going on that surpasses any machine. We can program a machine to believe that the consistency of PA is a hard fact. It would believe it without the need for proof, just as we do. >It means that no matter how long and in what order we keep generating >theorems we will never derive P & ~P. PA can be literally materialized >as a mechanical system, a machine. We are asking if tis machine can >ever produce P & ~P. No machine can answer that question. That's ridiculous. Of course a machine can answer that question. You can program a machine to answer "yes" when asked "Is PA consistent?". You can program a machine so that if asked why it believes that, it will say "Because the axioms are all manifestly true, and no contradiction can follow from manifestly true axioms." What is it you are saying cannot be done by any machine? -- Daryl McCullough Ithaca, NY > Newberry says... > [quoted text clipped - 19 lines] > can follow from manifestly true axioms." What is it you are saying cannot > be done by any machine? We can as well program a machine to say that PA is inconsistent or that it does not know. Is it a pure accident that PA is consistent and we were also programmed to believe that PA is consistent? Are you saying that we were purely arbitrarily programmed to say "PA is consistent, because the axioms are all manifestly true, and no contradiction can follow from manifestly true axioms", and our psychological makeup was arbitrarily shaped such that the argument appears compelling to us? We could as well have been made such that "PA is consistent" would appear absurd to us? Is this what "psychological reason" means? Newberry says... >> That's ridiculous. Of course a machine can answer that question. You >> can program a machine to answer "yes" when asked "Is PA consistent?". [quoted text clipped - 5 lines] >We can as well program a machine to say that PA is inconsistent or >that it does not know. Right. For any response that a human could give, we can program a machine to give the same response. So it is false that human abilities surpass any machine. >Is it a pure accident that PA is consistent and >we were also programmed to believe that PA is consistent? No, it's not an accident. PA is a generalization and systematization of human experience with natural numbers over thousands of years. It's a very natural thing to come up with if one wants to formalize mathematics. A machine with heuristics for coming up with axiomatizations of mathematics would come up with the same axioms. >Are you saying that we were purely arbitrarily programmed to >say "PA is consistent, because the axioms are all manifestly true, and no >contradiction can follow from manifestly true axioms", and our >psychological makeup was arbitrarily shaped such that the argument >appears compelling to us? No, I'm not saying that. I'm saying that there is no reason to believe that a machine could not understand the consistency of PA in the same way humans do. Certainly Godel's theorem doesn't suggest otherwise. >We could as well have been made such that >"PA is consistent" would appear absurd to us? Is this what >"psychological reason" means? No, I'm saying exactly the opposite. There are good reasons for humans to believe PA is consistent, and a machine can perfectly well be programmed to believe in the same way. -- Daryl McCullough Ithaca, NY On Dec 10, 6:26 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 18 lines] > systematization of human experience with natural numbers > over thousands of years. Are you saying that mathematics is an empirical science? It's a very natural thing to > come up with if one wants to formalize mathematics. > A machine with heuristics for coming up with axiomatizations [quoted text clipped - 10 lines] > of PA in the same way humans do. Certainly Godel's theorem > doesn't suggest otherwise. You sound like a broken record. > >We could as well have been made such that > >"PA is consistent" would appear absurd to us? Is this what [quoted text clipped - 8 lines] > Daryl McCullough > Ithaca, NY Newberry says... >Are you saying that mathematics is an empirical science? There are two aspects of mathematics: (1) Deciding what mathematical principles are useful, interesting, consistent, etc. (2) Deriving what follows from those principles. The first aspect is certainly empirical. The second aspect is definitely not empirical, but it is something that we can easily program a machine to do. >> No, I'm not saying that. I'm saying that there is no reason >> to believe that a machine could not understand the consistency >> of PA in the same way humans do. Certainly Godel's theorem >> doesn't suggest otherwise. > >You sound like a broken record. Well, you keep saying the same false things over and over, so I feel inclined to correct them by saying the truth over and over. If it's boring, then stop. Stop saying false things, or at least give more of an argument as to why you believe them. Put it in the form of a syllogism: Give your starting assumptions, and show how your conclusion is supposed to follow. Unfortunately, you always leave out the crucial step: Given: (1) Humans believe that PA is consistent. (2) The only proofs of the consistency of PA rely on mathematical principles that go beyond PA. (3) The only proofs of the consistency of those principles rely on yet other principles. (4) etc. I grant all of that. But the conclusion "Therefore, human abilities surpass those of any machine" does *not* follow from those facts. Do you know what it means for one statement to follow from other statements? At least in one characterization of it, a statement C follows from hypotheses H1, H2, etc. if there is a sequence of statements S_1, S_2, ... such that each S_j is either a hypothesis, or it is a theorem of pure predicate logic (no nonlogical axioms), or it follows from earlier statements by logical deduction. Do you actually think that the conclusion "Human abilities surpass those of any machine" follows in this sense from Godel's incompleteness theorem, together with the observation that humans believe that PA is consistent? If so, demonstrate it. -- Daryl McCullough Ithaca, NY -- Daryl McCullough Ithaca, NY On Dec 11, 6:18 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 52 lines] > humans believe that PA is consistent? If so, > demonstrate it. No, I believe that this follows; either 1) We do not know if PA is consistent (I mean we know zilch, we do not even have a good reason to presume that it is consistent.) 2) The human mind can perform non-computable functions 3) There exists a formalization of arithmetic that can prove its own consistency. (That would explain our intuition that PA is consistent.) I did put my argument in syllogism. a) Humans are certain that PA is consistent b) No formal proof of PA has any cogency (TF explicitly admitted this) c) Machines are capable only of formal proofs d) Hence humans can perform feats no computer can But you chose to ignore it. > On Dec 11, 6:18 am, stevendaryl3...@yahoo.com (Daryl McCullough) > wrote: [quoted text clipped - 69 lines] > d) Hence humans can perform feats no computer can > But you chose to ignore it. c) is clearly wrong. A clock is capable of giving the time - that's not a formal proof. A printer is capable of printing - that's not a formal proof. Look, maybe we are just machines and God programmed us at the beginning. Babies are machines that their parents program. Your whole argument obviously needs to be clear about what is or is not a machine. You haven't done this; so it's not going to get anywhere. Newberry says... >On Dec 11, 6:18 am, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> Do you actually think that the conclusion >> "Human abilities surpass those of any machine" [quoted text clipped - 9 lines] >3) There exists a formalization of arithmetic that can prove its own >consistency. (That would explain our intuition that PA is consistent.) No, those *DON'T* follow. For one thing, Godel's theorem is about proof, while the claim "We know that PA is consistent" or "We don't know that PA is consistent" is about *knowledge*. What are you assuming about the relationship between knowledge and proof? >I did put my argument in syllogism. >a) Humans are certain that PA is consistent Okay, we grant that as an assumption. >b) No formal proof of PA has any cogency (TF explicitly admitted this) I don't grant that at all. Why not say what I just said: There is no proof of the consistency of PA except for those use principles that go beyond PA. >c) Machines are capable only of formal proofs That's a ridiculous thing to say. That's completely false. I can program a machine to play games, process images, process sound, solve differential equations, etc. That's not proving formal proofs. >d) Hence humans can perform feats no computer can >But you chose to ignore it. Because it is a ridiculous syllogism. It's nonsense. For one thing, it isn't a syllogism at all, since the conclusions don't logically follow from the premises. For another thing, it relies on manifestly false premises. You haven't spelled your assumptions about what connection there is between knowledge and proof. What is the connection? You haven't said what you mean by "Machines are only capable of proof". What do you think is the relationship between being certain and proof? Why do you insist that a machine must have a proof to be certain, but that a human does not need a proof? You're using a double standard. -- Daryl McCullough Ithaca, NY On Dec 12, 6:34 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 19 lines] > What are you assuming about the relationship between knowledge > and proof? I mean thta knowledge is a superset of proofs. hence the human mins surpasses any machine (if you exclude 1 and 3.) > >I did put my argument in syllogism. > >a) Humans are certain that PA is consistent [quoted text clipped - 6 lines] > is no proof of the consistency of PA except for those use principles > that go beyond PA. I thought we already sttles this one. I am amazed that people claim that when a system proves its own consistency the proof does not have any cogency but when you prove it is in a stronger system it does. > >c) Machines are capable only of formal proofs > > That's a ridiculous thing to say. That's completely > false. I can program a machine to play games, process > images, process sound, solve differential equations, etc. > That's not proving formal proofs. Come on! Obviously I meant that machines cannot arrive at arithmetic result by any other means than formal proofs. > >d) Hence humans can perform feats no computer can > >But you chose to ignore it. [quoted text clipped - 12 lines] > must have a proof to be certain, but that a human > does not need a proof? The relationship is that we are certain about more things than we can prove. It is generally accepted that the set of true sentences of PA is a superset of the derivable sentences of PA. Here is a example: PA is consistent a) We have the knowldge that it is true, we are certain it is the case b) There does not exist any proof that can prove that PA is consistent. Of course, b) needs to be qualified but we went through that a few times and you yourself affirmed it. I think you will have to admit that either 1), 2), or 3) is the case. > You're using a double standard. THIS is ridiculous! Newberry says... >On Dec 12, 6:34 am, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> >No, I believe that this follows; either >> >1) We do not know if PA is consistent (I mean we know zilch, we do not [quoted text clipped - 10 lines] > >I mean that knowledge is a superset of proofs. Proofs from what theory? That doesn't make any sense. Every statement can be proved in *some* theory. >hence the human mind surpasses any machine (if you exclude 1 and 3.) Don't say "hence". It doesn't follow. Once again, *PLEASE* put it in the form of a syllogism. Every statement should (in principle, at least) follow from previously establish statements or assumptions, plus the rules of logic. If your terminology changes from statement to statement (for example, changing from "proof" to "knowledge") you need additional bridging statements showing what you are assuming about the connection between these terms. >> >I did put my argument in syllogism. >> >a) Humans are certain that PA is consistent [quoted text clipped - 8 lines] > >I thought we already sttles this one. No, we did not. >I am amazed that people claim that when a system proves its own >consistency the proof does not have any cogency but when you prove >it is in a stronger system it does. Whether a proof is "cogent" is a *psychological* fact. Are we convinced by it? There are convincing arguments for the consistency of PA. They convince *me*, anyway. >> >c) Machines are capable only of formal proofs >> [quoted text clipped - 5 lines] >Come on! Obviously I meant that machines cannot arrive at arithmetic >result by any other means than formal proofs. What reason is there to say that? That's ridiculous. That's completely false. Almost *no* computer programs work by making formal proofs. Yet there are computer programs that arrive at mathematical conclusions. What you are saying is nonsense. -- Daryl McCullough Ithaca, NY On Dec 12, 7:57 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 42 lines] > > No, we did not. You admitted this a few times on this thread. Torkel Franzen says "the soundness proof of PA is not intended to allay any doubts at all." p. 110 How do you know that ZFC + an axiom of infinity are consistent? > >I am amazed that people claim that when a system proves its own > >consistency the proof does not have any cogency but when you prove [quoted text clipped - 3 lines] > we convinced by it? There are convincing arguments for the > consistency of PA. They convince *me*, anyway. So you would not be convinced by a system that proves its own consistency but you would be convinced if were proven by principles that go beyond it? > >> >c) Machines are capable only of formal proofs > [quoted text clipped - 10 lines] > making formal proofs. Yet there are computer programs that > arrive at mathematical conclusions. Examples? > What you are saying is nonsense. So let's try the proof this way. Let us assume we are programmed as ZFC with a strong axiom of infinity. So we can prove the consistency of ZFC to ourselves. But we also know that we do not know if ZFC with a strong axiom of infinity is consistent. If it is not than the proof of ZFC's consistency is invalid. So we are convinced about ZFC's consistency because ZFC with a strong axiom of infinity is our horizont. We are fooling ourselves. If do not want to admit that we are fools we have to drop the assumption that we are equivalent to ZFC with a strong axiom of infinity. Let PA = T_0, ZFC = T_1, ZFC + axiom of infinity = T_2. There probably exists a theory T3 in which the consistency with a strong axiom of infinity can be proved. The argument in the first paragraph applies to any theory T_n for n > 1. That is are not equivalent to any theory T_n. How else can we possibly prove a consistency of a given theory T? A theory could prove its own consistency. That is our option 3). I do not know if a consistency of any theory T can be proven in a weaker system but if you want we can add option 4.) What else? A heuristic learning program cannot do it because mathematics is not an empirical science. A game playing program cannot do it because it doesn not even attempt to answer any arithmetic questions. That a system is consistent can be proven only in some theory. So in order to produce a consistency proof the computer has to programmed as a theory. So we have A) an inreasing hierarchy of theories,and we have proven in the first paragraph that either we are NOT equivalent to any of those or that we do not know if PA is consistent. B) theories proving their own consistencies C) maybe a decreasing hierarchy of theories. > > >I am amazed that people claim that when a system proves its own > > >consistency the proof does not have any cogency but when you prove > > >it is in a stronger system it does. Tertium datur, of course. You may be able to prove the consistency of one theory in another theory which is neither strictly weaker nor strictly stronger than the other. This is what happens with Gentzen's proof of the consistency of first-order PA using PRA + transfinite induction up to epsilon_0 for quantifier free formulae. If your worry about PA is that it has "too much" induction in allowing induction for arbitrarily complex formulae, then this proof could calm those worries by restricting induction to quantifier free formulae (though allowing "longer" inductions). > > > >I am amazed that people claim that when a system proves its own > > > >consistency the proof does not have any cogency but when you prove [quoted text clipped - 5 lines] > proof of the consistency of first-order PA using PRA + transfinite > induction up to epsilon_0 for quantifier free formulae. So noted. I also note that most people do not find Gentzen's proof very convincing. And I still do not see how a proof in a stronger system could be more convincing than a proof in the system itself. If your worry > about PA is that it has "too much" induction in allowing induction for > arbitrarily complex formulae, then this proof could calm those worries > by restricting induction to quantifier free formulae (though allowing > "longer" inductions). >So noted. I also note that most people do not find Gentzen's proof >very convincing. Is that really true? Are you basing this claim on statistical evidence? >And I still do not see how a proof in a stronger >system could be more convincing than a proof in the system itself. Correct me if I'm wrong, but I don't believe that PRA + induction up to epsilon_0 is "stronger" than PA, is it? "Stronger" meaning that it proves all theorems of PA. SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences On Dec 16, 10:46 am, tc...@lsa.umich.edu wrote: > In article <f1433411-6354-4ff3-9824-0b95ceb4d...@l32g2000hse.googlegroups.com>, > [quoted text clipped - 8 lines] > Correct me if I'm wrong, but I don't believe that PRA + induction up to > epsilon_0 is "stronger" than PA, is it? No, that's why it is a tertium. "Stronger" meaning that it proves > all theorems of PA. >On Dec 16, 10:46 am, tc...@lsa.umich.edu wrote: >> Correct me if I'm wrong, but I don't believe that PRA + induction up to >> epsilon_0 is "stronger" than PA, is it? > >No, that's why it is a tertium. What do you mean, "that's why it is a tertium"? I don't understand. SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences Newberry says... >So noted. I also note that most people do not find Gentzen's proof >very convincing. And I still do not see how a proof in a stronger >system could be more convincing than a proof in the system itself. The *strength* of the system is not relevant so much as whether the axioms are themselves intuitively true. A proof in a theory whose axioms are intuitively true is more useful and interesting than a proof in a theory whose axioms are not intuitively true. So, for example, a proof in PA + the negation of Goldbach's conjecture would not be very convincing, because we have no reason to believe that the negation of Goldbach's conjecture is true. -- Daryl McCullough Ithaca, NY > Newberry says... > [quoted text clipped - 11 lines] > reason to believe that the negation of Goldbach's conjecture > is true. Unfortunately mathematical reasoning isn't religion where "beliefs" would be much relevant. > -- > Daryl McCullough > Ithaca, NY >Unfortunately mathematical reasoning isn't religion where "beliefs" >would be much relevant. You don't think beliefs are relevant in mathematical reasoning? Then how do you become convinced that *anything* is true? Are you convinced, for example, that sqrt(2) is irrational? On what basis? On the basis of the proof? But the proof starts with some axioms. On what basis do you become convinced of the correctness of the axioms? Or are you *not* convinced of the axioms? But if you're not convinced of the axioms, then what good is a proof of "sqrt(2) is irrational" from those axioms? SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences >> Unfortunately mathematical reasoning isn't religion where "beliefs" >> would be much relevant. > > You don't think beliefs are relevant in mathematical reasoning? From what I gather, we don't call that "beliefs". We call it _interpretation_ which model basically is, and in which truths are true or false. The problem of this model-truth is over the same "structure" there could be opposite interpretation. Religious truth on the other hand is supposed to *believed* as true whether or not there is a model to reflect the truth. That's why belief doesn't have much of relevance in reasoning. > Then how do you become convinced that *anything* is true? As I've explained above. > Are you convinced, for example, that sqrt(2) is irrational? On what basis? On the basis of model that "sqrt(2) is irrational" is true, of course. > On the basis of the proof? No, not on the basis of proof: what is true or false is based strictly on model. Syntactical provability is actually in a different (and independent) paradigm, not withstanding Completeness. > But the proof starts with some axioms. Of course. > On what basis do you become convinced of the correctness of the axioms? What exactly does "correctness of the axioms" mean? > Or are you *not* convinced of the axioms? The only senses for which we could talk about axioms are: (a) They be independent from each other. (b) They don't contradict each other. So, again, what does it mean to be "convinced of the axioms"? > But if you're not convinced of the axioms, then what good is a proof of > "sqrt(2) is irrational" from those axioms? Proofs of course are good as a mechanism of assisting us in preventing our reasoning from being inconsistent. Of course. > tc...@lsa.umich.edu wrote: > From what I gather, we don't call that "beliefs". We call it _interpretation That we have a method for interpreting formal languages does not contradict that one may believe certain things about mathematics. > which model basically is, and in which truths are true or false. "in which truths are true or false". Are you sure you've eaten breakfast this morning? > The problem > of this model-truth is over the same "structure" there could be opposite > interpretation. No, that's completely wrong. Given a structure, there is only one interpretation associated with that structure. > Religious truth on the other hand is supposed to *believed* > as true whether or not there is a model to reflect the truth. I've never seen such a description of religious belief. > That's why > belief doesn't have much of relevance in reasoning. Belief may or may not have relevance in reasoning, but the confusions you just posted don't lead to any conclusion on the matter. > > Then how do you become convinced that *anything* is true? > > As I've explained above. No you didn't. > > Are you convinced, for example, that sqrt(2) is irrational? On what basis? > > On the basis of model that "sqrt(2) is irrational" is true, of course. Maybe you mean, on the basis that there is a model in which "sqrt(2) is irrational" is true. And there is a model in which it is false also. What about operations on finite strings? Don't you believe, for example, irrespective of any model, that the string "0011" is the same as the string "0022 [with 1 substituted for 2]"? > > On the basis of the proof? > [quoted text clipped - 16 lines] > (a) They be independent from each other. > (b) They don't contradict each other. No, there are lots of other properties of axioms. One, for example, is that of a certain model being a model of the axioms. > So, again, what does it mean to be "convinced of the axioms"? > [quoted text clipped - 3 lines] > Proofs of course are good as a mechanism of assisting us in preventing > our reasoning from being inconsistent. Of course. Except if the axioms are inconsistent. Actually, (first order) proof doesn't ensure consistency but rather entailment. MoeBlee >> tc...@lsa.umich.edu wrote: > [quoted text clipped - 7 lines] > "in which truths are true or false". Are you sure you've eaten > breakfast this morning? I usually have a glass of milk every morning. Some *interpret* that as having breakfast; others would have opposite interpretation. Actually my interpretation on that might vary from years to year. What's about yours interpretation on this? >> The problem >> of this model-truth is over the same "structure" there could be opposite >> interpretation. > > No, that's completely wrong. Given a structure, there is only one > interpretation associated with that structure. Given the structure "Nam's having a glass of milk every morning", how many interpretations would one have for the sentence "Nam has breakfast every morning"? >> Religious truth on the other hand is supposed to *believed* >> as true whether or not there is a model to reflect the truth. > > I've never seen such a description of religious belief. I'm sure there are descriptions that are very much *similar*! >> That's why >> belief doesn't have much of relevance in reasoning. > > Belief may or may not have relevance in reasoning, but the confusions > you just posted don't lead to any conclusion on the matter. Whose "confusions" are you talking about? I don't seem to have any here. >>> Then how do you become convinced that *anything* is true? >> As I've explained above. > > No you didn't. That's one opinion of course. >>> Are you convinced, for example, that sqrt(2) is irrational? On what basis? >> On the basis of model that "sqrt(2) is irrational" is true, of course. > > Maybe you mean, on the basis that there is a model in which "sqrt(2) > is irrational" is true. "Maybe"? My answer to Tim Chow's question is a straightforward short-one-liner answer and you seemed to not understand? > And there is a model in which it is false also. So far I don't see what your point here is! > What about operations on finite strings? What about them? > Don't you believe, for > example, irrespective of any model, that the string "0011" is the same > as the string "0022 [with 1 substituted for 2]"? In what context are you talking about "sameness", "substitute", etc... Sorry your question is too vague in semantic and consequently is subject to different interpretations. >>> On the basis of the proof? >> No, not on the basis of proof: what is true or false is based strictly on model. [quoted text clipped - 15 lines] > No, there are lots of other properties of axioms. One, for example, is > that of a certain model being a model of the axioms. Of course there are other properties: axioms' being finite formulas, etc... All of these (and what you've mentioned) are utterly trivial not worth being mentioned, it seems. So why are you mentioning here? Is it because it has something to do with the purported "correctness of the axioms" that is being discussed between me and the other poster? (Besides, to be to be a property of axioms it has to apply to all axioms in all circumstances, e.g. being finite formulas. Your "certain model being a model of the axioms" is not applicable to all axioms!) >> So, again, what does it mean to be "convinced of the axioms"? >> [quoted text clipped - 5 lines] > Except if the axioms are inconsistent. Actually, (first order) proof > doesn't ensure consistency but rather entailment. Agree. Except that I only said "as a mechanism of assisting": I never said anything about proofs guaranteeing/ensuring consistency. Of course not, in general. But in some particular circumstances, proof would help consistent reasoning. > MoeBlee > I usually have a glass of milk every morning. Some *interpret* that > as having breakfast; others would have opposite interpretation. > Actually my interpretation on that might vary from years to year. > What's about yours interpretation on this? That your postings make apparent that for you a class of milk is not sufficient nutrition to maintain a healthy working brain. > >> The problem > >> of this model-truth is over the same "structure" there could be opposite [quoted text clipped - 6 lines] > many interpretations would one have for the sentence "Nam has breakfast > every morning"? It doesn't matter, as long as you understand that given a structure, there is only one interpretation associated with that structure, and that your statement "over the same "structure" there could be opposite interpretation" is nonsense. > >> Religious truth on the other hand is supposed to *believed* > >> as true whether or not there is a model to reflect the truth. > > > I've never seen such a description of religious belief. > > I'm sure there are descriptions that are very much *similar*! I've never seen one. I'm not an expert on religion though, so I welcome any example you wish to adduce. > >> That's why > >> belief doesn't have much of relevance in reasoning. [quoted text clipped - 3 lines] > > Whose "confusions" are you talking about? I don't seem to have any here. You're still tragically confused about what a model is, as I already mentioned in my previous post.. > >>> Then how do you become convinced that *anything* is true? > >> As I've explained above. > > > No you didn't. > > That's one opinion of course. And it's your opinion to the contrary. Gee whiz, thanks for bringing us to that stunning realization. > >>> Are you convinced, for example, that sqrt(2) is irrational? On what basis? > >> On the basis of model that "sqrt(2) is irrational" is true, of course. [quoted text clipped - 4 lines] > "Maybe"? My answer to Tim Chow's question is a straightforward short-one-liner > answer and you seemed to not understand? Your answer is syntactically garbled. I'm just asking whether I've correctly understood you despite your garbled syntax. > > And there is a model in which it is false also. > > So far I don't see what your point here is! That you haven't accounted for any sense in which "sqrt(2) is irrational" is true given that it is also false. > > What about operations on finite strings? > > What about them? I mention this below. Obviously, you're typing your responses one line at a time, like a real wise-a.s, not bothering to read on for the rest of the message. I've long said that you are an obnoxious twit. > > Don't you believe, for > > example, irrespective of any model, that the string "0011" is the same [quoted text clipped - 3 lines] > Sorry your question is too vague in semantic and consequently is subject > to different interpretations. Just the plain ordinary sense in which you recognize a foramula as it is conveyed by printed text or by whatever means of conveyence. Do you have any notion at all of finitistic operations on strings of symbols? > >>> On the basis of the proof? > >> No, not on the basis of proof: what is true or false is based strictly on model. [quoted text clipped - 19 lines] > All of these (and what you've mentioned) are utterly trivial not worth > being mentioned, it seems. So why are you mentioning here? Because they're not trivial, and because they are counterexamples to your over-generalization. > Is it because > it has something to do with the purported "correctness of the axioms" that [quoted text clipped - 3 lines] > in all circumstances, e.g. being finite formulas. Your "certain model being > a model of the axioms" is not applicable to all axioms!) Oh please, how mindlessly pedantic can you get. Obviously, I'm speaking in the same sense in which I would by saying "evenness is a property of integers" where it is meant not that every integer is even but rather that integers are even or not, whereas for certain other kinds of objects evenness is not even at issue. > >> So, again, what does it mean to be "convinced of the axioms"? > [quoted text clipped - 9 lines] > anything about proofs guaranteeing/ensuring consistency. Of course not, in general. > But in some particular circumstances, proof would help consistent reasoning. Okay. MoeBlee >> I usually have a glass of milk every morning. Some *interpret* that >> as having breakfast; others would have opposite interpretation. [quoted text clipped - 3 lines] > That your postings make apparent that for you a class of milk is not > sufficient nutrition to maintain a healthy working brain. Of course talking about mathematical reasoning with you seems fruitless, based on many of your posts in the past: they're seem to be more about "attacking", "character assassination", or what have we, and less about the *genuine discussing* about underlying points of contention. So hope you don't mind my stopping dialog with you at this point in time. > MoeBlee > >> I usually have a glass of milk every morning. Some *interpret* that > >> as having breakfast; others would have opposite interpretation. [quoted text clipped - 5 lines] > > Of course talking about mathematical reasoning with you seems fruitless, It's fruitless with you because you cling tenaciously to your misconceptions about even basic concepts such as that of a model. > based on many of your posts in the past: they're seem to be more about "attacking", > "character assassination", No, they're not MORE about commenting on you personally. I engage (and have engaged extensively) on substantive matters with you. That I ALSO remark about you personally doesn't erase my substantive comments. MoeBlee > >>> Are you convinced, for example, that sqrt(2) is irrational? On what basis? > [quoted text clipped - 21 lines] > Sorry your question is too vague in semantic and consequently is subject > to different interpretations. OK, I'll bite: Yes, the string "0011" is the same as the string "0022 [with 1 substituted for 2]". What does that have to do with models and "truth" and "sqrt(2) is irrational" etc.? -- hz >>>>> Are you convinced, for example, that sqrt(2) is irrational? On what basis? >>>> On the basis of model that "sqrt(2) is irrational" is true, of course. [quoted text clipped - 19 lines] > "0022 [with 1 substituted for 2]". What does that have to do > with models and "truth" and "sqrt(2) is irrational" etc.? Good question. Apparently MoeBlee (not I) is the one who asked the question about some "string". > -- > hz > >>>>> Are you convinced, for example, that sqrt(2) is irrational? On what basis? > >>>> On the basis of model that "sqrt(2) is irrational" is true, of course. [quoted text clipped - 22 lines] > Good question. Apparently MoeBlee (not I) is the one who asked the question > about some "string". It has to do with the bases upon which one has certain mathematical beliefs. Does one need a notion of model to believe that the two strings are the same? (And sameness of strings is a mathematical subject). MoeBlee > > >>> Are you convinced, for example, that sqrt(2) is irrational? On what basis? > [quoted text clipped - 25 lines] > "0022 [with 1 substituted for 2]". What does that have to do > with models and "truth" and "sqrt(2) is irrational" etc.? It has to do with the bases upon which one has certain mathematical beliefs. Does one need a notion of model to believe that the two strings are the same? (And sameness of strings is a mathematical subject). MoeBlee >> Are you convinced, for example, that sqrt(2) is irrational? On what basis? > >On the basis of model that "sqrt(2) is irrational" is true, of course. You didn't answer my first question directly. Are you convinced that sqrt(2) is irrational? Or are you convinced only of the statement, "in the standard model of the integers, there are no integers m and n such that m^2 = 2 n^2"? Let me assume that the latter formulation is the only one that you assent to. Then let me ask this: Do you believe (note the word "believe" here) the following statement? (*) In the standard model of the integers, there are no integers m and n such that m^2 = 2 n^2. If so, on what basis do you believe (*)? Not on the basis of proof, according to what you say later. So what other basis do you have? SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences > (*) In the standard model of the integers, there are no integers m and n > such that m^2 = 2 n^2. I meant "no nonzero integers m and n," of course. SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences >>> Are you convinced, for example, that sqrt(2) is irrational? On what basis? >> On the basis of model that "sqrt(2) is irrational" is true, of course. > > You didn't answer my first question directly. Are you convinced that sqrt(2) > is irrational? But your first question is unanswerable, *without* a context/basis! (Why did you ask "On what basis?" *immediately* after your first question?) If you asked me whether or not the sentence "President Kennedy is dead" is true/false, I'd react in the same way: no context no possible answer. If this sentence is answered in a play tomorrow about an event in 1960, the sentence could be false, in that context. If the play happens to be actually in 1961, but about an imagined event in 2010 the sentence could still be true or false, depending on the context of the plot the play-writer intended to write. If this sentence is uttered in the "normal" (i.e. historical) context, there are still other contexts to be content with: in some religion "beliefs" there would be no dead, just one life after another; and accordingly, President Kennedy would be still "alive", years after ... well, his "assassination"! Philosophical "crab" right? Not really! Especially when talking about something that looks like a FOL truth question, like is "sqrt(2) is irrational". Let's even deal with a much simpler question: is "1+1=0" true? The way DMC, PS and you seem to say is that there is certain *absolute* truth (value) about "1+1=0" we must necessarily "believe" in, in much the same way some "religiously believe" Jesus of Nazareth is only son of God. What I counter is that there's no such kind of unchangeable belief in mathematical reasoning. What would be a distinction between these 2 kinds of belief: for one thing, model-truth belief (which you seemed to allude below) is changeable/relative/subjective; given the same event, structure, sets of ... (or whatever we may want to call), we are at the liberty to change that belief 180 degree and still are correct as far as mathematical "belief" viz-a-viz reasoning is concerned. That's not the same kind of religion beliefs: there its truth values are absolute, immutable, independent of any human perception, or even ... well, belief! > Or are you convinced only of the statement, "in the standard > model of the integers, there are no integers m and n such that m^2 = 2 n^2"? There's a misconception here that seems to have escaped your attention. A model *always already* includes a *chosen* interpretation: hence a belief has been "believed" already. What's important is this interpretation could always be reversed to the other way - at will - and a opposite "belief" would occur. Consequently, a mathematically *stated* belief could change back and forth, as in the following "The Lady's Yes" poem: "Yes," I answered you last night; "No," this morning, Sir, I say. Colours seen by candlelight, Will not look the same by day. [...] In summary, for "there are no [nonzero] integers m and n such that m^2 = 2 n^2", its truth, or its believed truth is quite subjective and relative, like the "Colours seen by candlelight", but unlike religion truth and belief. Which is my whole point here. > Let me assume that the latter formulation is the only one that you assent to. > Then let me ask this: Do you believe (note the word "believe" here) the [quoted text clipped - 5 lines] > If so, on what basis do you believe (*)? Not on the basis of proof, according > to what you say later. So what other basis do you have? > A model > *always already* includes a *chosen* interpretation: hence a belief has been > "believed" already. A model is a mathematical object. I don't know how you would argue that a model requires "a belief has been "believed" already". MoeBlee >> A model >> *always already* includes a *chosen* interpretation: hence a belief has been >> "believed" already. > > A model is a mathematical object. To be precise, it's more than just a mere mathematical object: it's an *interpreted* mathematical object. A syntactical wff on the other hand is a non-interpreted mathematical object. What is the difference? Well, the entire Gestalt view of the collection of components of a wff is supposed to be independent of any being's interpreation or view: a FOL formufla would mean the same to all - no choice of an alternative. The Gestalt view of the collection of components of a model-structure is a model basically, and is subject to individual reasoner's interpretation. Given the same structure, you could subjectively interpret the Gestalt view differently! For instance, given the following structure: xxxbxxxxxxxxxxxxxxxxxxxxxxxxxx Would you *interpret* that structure as a model of the blue-eye-dragon theory T? Or would that be *not* a model of T, because you would interpret "b" as "non-blue" (e.g. "brown"), and in such case it could be a model of a non-blue-eye-dragon theory T' (if you so care to view it that way). On the other hand, given an appropriate language, the formula F df= "the dragon has a blue eye" could not be viewed/interpreted differently. For instance, that formula could not be viewed as ~F. > I don't know how you would argue > that a model requires "a belief has been "believed" already". Just change "a belief" to "an interpretation" and "believed" to "interpreted" then you would know. > MoeBlee > >> A model > >> *always already* includes a *chosen* interpretation: hence a belief has been [quoted text clipped - 3 lines] > > To be precise, it's more than just a mere mathematical object: It's not just a mathematical object, in the sense that it is a certain kind of mathematical object (as any mathematical object is a certain kind of mathematical object). > it's an *interpreted* > mathematical object. I'd say it IS an interpretation (or, depending on the definition, one of its components is an interpretation). > A syntactical wff on the other hand is a non-interpreted > mathematical object. A wff is not ordinarily itself an interpretation, okay. > What is the difference? Well, the entire Gestalt view of the > collection of components of a wff is supposed to be independent of any being's [quoted text clipped - 3 lines] > Given the same structure, you could subjectively interpret the Gestalt view > differently! I'll leave that to you, but I'll read on... > For instance, given the following structure: > > xxxbxxxxxxxxxxxxxxxxxxxxxxxxxx You're using 'structure' here in what sense exactly? > Would you *interpret* that structure as a model of the blue-eye-dragon > theory T? Or would that be *not* a model of T, because you would interpret [quoted text clipped - 3 lines] > has a blue eye" could not be viewed/interpreted differently. For instance, > that formula could not be viewed as ~F. I have no idea what you're talking about. Sorry. > > I don't know how you would argue > > that a model requires "a belief has been "believed" already". > > Just change "a belief" to "an interpretation" and "believed" to "interpreted" > then you would know. Then I get: A model requires an interpretation has been interpreted. Sorry, I have no idea what motivates you to string those words together. If it is of any help, what I already said: A model IS an interpretation (or, depending on the definition, includes an interpretation as a component. MoeBlee P.S. Possibly a response later by you to my example about string substitution? >There's a misconception here that seems to have escaped your attention. A model >*always already* includes a *chosen* interpretation: hence a belief has been >"believed" already. What's important is this interpretation could always be >reversed to the other way - at will - and a opposite "belief" would occur. >Consequently, a mathematically *stated* belief could change back and forth, [...] >In summary, for "there are no [nonzero] integers m and n such that >m^2 = 2 n^2", its truth, or its believed truth is quite subjective and >relative, like the "Colours seen by candlelight", but unlike religion >truth and belief. So let's look at these two statements: (1) There are no nonzero integers m and n such that m^2 = 2 n^2. (2) In the standard model of the integers, there are no nonzero integers m and n such that m^2 = 2 n^2. As I understand your position, (1) does not have a determinate truth value. But in (2), the phrase "in the standard model of the integers" chooses an interpretation and hence a "belief has been `believed' already." Does that mean that (2) is absolutely, unconditionally true? *After* I choose an interpretation, the subjectivism and relativism are gone, aren't they? SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences >> There's a misconception here that seems to have escaped your attention. A model >> *always already* includes a *chosen* interpretation: hence a belief has been [quoted text clipped - 17 lines] > But in (2), the phrase "in the standard model of the integers" chooses > an interpretation and hence a "belief has been `believed' already." > Does that mean that (2) is absolutely, unconditionally true? No. What is a standard model of the integers might not be the "standard" model to others! > *After* I choose an interpretation, the subjectivism and relativism > are gone, aren't they? No, they're just there besides you. That the speed of a train is absolutely 100 km/hr from you point of view doesn't make the speed of train be an absolute quantity! >>> There's a misconception here that seems to have escaped your >>> attention. A model [quoted text clipped - 29 lines] > No. What is a standard model of the integers might not be the "standard" > model to others! Let me re-phrase it: what might be "integers" or "standard" to one, might not be to the others. >> *After* I choose an interpretation, the subjectivism and relativism >> are gone, aren't they? [quoted text clipped - 3 lines] > absolute > quantity! >> (1) There are no nonzero integers m and n such that m^2 = 2 n^2. >> >> (2) In the standard model of the integers, there are no nonzero integers >> m and n such that m^2 = 2 n^2. [...] >> Does that mean that (2) is absolutely, unconditionally true? > >No. What is a standard model of the integers might not be the "standard" >model to others! All right, then, let's try this one: (3) In every model of PA, there are no nonzero integers m and n such that m^2 = 2 n^2. Most people would agree with (3), since the proof of the irrationality of sqrt(2) can be formalized in PA, and therefore the statement holds in all models of PA, standard or nonstandard. Do you believe (3)? Is (3) absolutely true? Whether or not your "standard" model is the same as mine makes no difference, since the assertion holds in every model, so I don't see where relativism enters. SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences >>> (1) There are no nonzero integers m and n such that m^2 = 2 n^2. >>> [quoted text clipped - 17 lines] > model is the same as mine makes no difference, since the assertion holds in > every model, so I don't see where relativism enters. Apparently you've missed my last post where I made the correction: > Let me re-phrase it: what might be "integers" or "standard" to one, > might not be to the others. Any rate, so (3) is true, *relative* to PA's models: the *truth* of (3) is a relative value, and the relativity is still there! > tc...@lsa.umich.edu wrote: > >> tc...@lsa.umich.edu wrote: [quoted text clipped - 27 lines] > Any rate, so (3) is true, *relative* to PA's models: the *truth* of (3) > is a relative value, and the relativity is still there! (1) No, you completely dodged the point. Okay, in a technical sense, '2=1+1' is true relative to models because it's true in some models but not in others. But the challenge put to you was to say in what way (3) is relative to models, since it's not a matter of being true in some models and not in others, but rather of being true PERIOD, since it is a statement about ALL models. To reinforce that it is a statement about all models, please recognize that it is of the form, Given ANY model, if it is a model of PA, then [...]. (2) What is the relativity in '0011' being the same string as '0022 with 1 substituted for 2'? MoeBlee > Okay, in a technical sense, '2=1+1' is true relative to models because > it's true in some models but not in others. No, true in some interpretations but not in others. Models, as OPPOSED to interpretations, have to be models OF something. In this case, we are talking about PA and models are models of PA. And 2=1+1, being a THEOREM of PA, is true in ALL models (of PA). So do NOT go around giving Nam any "Okay"s. Nam is an idiot. > > Okay, in a technical sense, '2=1+1' is true relative to models because > > it's true in some models but not in others. > > No, true in some interpretations but not in others. > Models, as OPPOSED to interpretations, have to be models > OF something. This terminological point has been discussed before in threads of not too distant past. The phrase 'true in a model' is routine and vastly understood. And though less routine, upon just a bit of explanation, it can be seen that it turns out (it is a theorem) that every structure for a language is a model (of some theory) and, of course, by definition, every model of a theory is a structure (for some language). Thus, theorem: M is a model iff M is a structure. This is derivable from defintions of these terms as given in Enderton's 'A Mathematical Introduction To Logic', though perhaps he does not make a point of mentioning it. Moreover, the use of the word 'interpretation' seems to to depend on the author (as do the particular definitions of 'model' and 'structure'). I haven't used the word 'interpretation' as technically defined. But some authors take the interpretation to be the function from the symbols to the object, or as a system of functions, or other arrangements. Meanwhile, I simply use 'structure' and 'model' with the precise definitions as in Enderton's book. And I distinguish between: a structure FOR a language and a model OF a theory which, given that every model is a structure, can be written by interchanging the words 'model' and 'structure' in any of the four possible combinations in the above. > In this case, we are talking about PA and > models are models of PA. A model of PA may be also a model of a different theory from PA. Of course, if we stipulate that we're talking only of models of PA, then '2=1=1' is true in those models. But my remark to Nam was not premised with the stipulation that we're talking only of models of PA. > And 2=1+1, being a THEOREM > of PA, is true in ALL models (of PA). Yes, of course. > So do NOT go around > giving Nam any "Okay"s. What I said is correct. The sentence '2=1+1" is true in some models and false in others, and there is nothing misleading in my saying that, especially in the rest of the context of my remark. > Nam is an idiot. He's got some conceptual blocks. You, on the other hand, are impossible in your own very special way. I just let go by uncontested thousands of words of yours, some of them HILARIOUSLY ill-conceived (I just love your 0 = {x | Ay ~yex}, which you posted to CORRECT a bunch of people who had ALREADY CORRECTLY observed various equations with 0 !!!) and also intellectually hypocritical (the line you're arguing about semantics and axioms lately is the EXACT NEGATION of the line you argued, rather by spraying your mouth-foam in my face, when we first exchanged posts), since I learned a while ago that not only is there no point trying to get through to you but doing so is an ESPECIALLY unpleasant endeavor. MoeBlee > I just love your 0 = {x | Ay ~yex} ... On the other hand, 0 = x <-> Ay ~yex is just fine. :-) [I'm sure that this is what he originally had in mind here. ;-)] F. SignatureE-mail: info<at>simple-line<dot>de > > I just love your 0 = {x | Ay ~yex} ... > [quoted text clipped - 5 lines] > > [I'm sure that this is what he originally had in mind here. ;-)] But what makes it egregious is that he was, in his usually charming way, storming in tell everybody else that they're wrong (they weren't), and even breathing fire about getting the definition wrong when it wasn't even a DEFINITION that was at stake. MoeBlee >> [I'm sure that this is what he originally had in mind here. ;-)] >> > But what makes it egregious is that he was, in his usually charming > way, storming in tell everybody else that they're wrong (they > weren't), and even breathing fire about getting the definition wrong > when it wasn't even a DEFINITION that was at stake. Yes, yes, we know good old george. :-) On the other hand, his posting manners have improved considerably in recent years. (Did you notice that?) F. SignatureE-mail: info<at>simple-line<dot>de > Yes, yes, we know good old george. :-) On the other hand, his posting > manners have improved considerably in recent years. I see no reason to think that. MoeBlee > > > I just love your 0 = {x | Ay ~yex} ... > [quoted text clipped - 5 lines] > > > [I'm sure that this is what he originally had in mind here. ;-)] IF EVERYBODY IS SO SURE OF that, then why I are you still harping on it? A lot of people ORIGINALLY LET THAT SLIDE. If one more (FF) had, I could've canceled it. That is simply a mistake. It is not relevant to anything. It shows nothing other than that people sometimes make mistakes. > But what makes it egregious is that he was, in his usually charming > way, storming in tell everybody else that they're wrong (they > weren't), They WERE SO TOO, they STILL ARE, and you are A COMPLETE a.shole for even TRYING this!! FF is a lot righter than you are about what I was trying to say about 0 and what all THAT really means, but HE was WRONG about the original reply to apoorv's use of unrestricted comprehension! What WAS RELEVANT was ZFC, and that ZFC DOES NOT HAVE unrestricted comprehension! > But what makes it egregious is that he was, in his usually charming > way, storming in tell everybody else that they're wrong (they > weren't), and even breathing fire about getting the definition wrong > when it wasn't even a DEFINITION that was at stake. > > MoeBlee > FF [...] HE was WRONG about the original reply to > apoorv's use of unrestricted comprehension! Actually, I didn't encourage apoorv in using "unrestricted comprehension". > What WAS RELEVANT was ZFC, and that ZFC DOES NOT HAVE > unrestricted comprehension! Right. Though Paul R. Halmos mentions the convention to write {x : phi(x)} _even in ZFC_, if we can show ExAy(y e x <-> phi(y)). Hence we may actually formulate the definition 0 =df {x : x =/= x} (using this convention). Of course, FIRST we have to show ExAy(y e x <-> y =/= y). And you KNOW that this _can_ be shown (in ZFC). F. SignatureE-mail: info<at>simple-line<dot>de > > > > I just love your 0 = {x | Ay ~yex} ... > [quoted text clipped - 11 lines] > That is simply a mistake. It is not relevant to anything. > It shows nothing other than that people sometimes make mistakes. Just to be clear, the remark responded to there is by the poster G. Frege, not by me. > > But what makes it egregious is that he was, in his usually charming > > way, storming in tell everybody else that they're wrong (they [quoted text clipped - 8 lines] > What WAS RELEVANT was ZFC, and that ZFC DOES NOT HAVE > unrestricted comprehension! That ZFC does not have unrestricted comprehension does not contradict that the formulas posted by the poster G. Frege and some other people were correct - are theorems of ZFC - and were not claimed to be definitions. > > But what makes it egregious is that he was, in his usually charming > > way, storming in tell everybody else that they're wrong (they > > weren't), and even breathing fire about getting the definition wrong > > when it wasn't even a DEFINITION that was at stake. Right. Just as you left my quote there intact, no one (barring any exception I don't recall) claimed to be giving a DEFINITION, and their formulas (not including the original one that people were responding to) were correct and you, as usual, made a bloody foaming fool of yourself. MoeBlee > That ZFC does not have unrestricted > comprehension does not contradict > that the formulas posted by the poster >G. Frege and some other people were correct It does so too. > - are theorems of ZFC - They INCLUDED TERMS which ZFC *does*NOT*construct*. They were as such not even grammatical within the context of ZFC. > and were not claimed to be definitions. Which is of course utter bullshit -- EVERY time you write a sentence (theorem or otherwise), EVERY closed term in it is a definition of its referent. > > > But what makes it egregious is that he was, in his usually charming > > > way, storming in tell everybody else that they're wrong (they > > > weren't), and even breathing fire about getting the definition wrong > > > when it wasn't even a DEFINITION that was at stake. The definition of the empty set WAS at stake in that thread. I am not going to belabor the point vs. anyone who chooses to deny this; anybody whose respect I care to care about will be able to see it for himself. > Right. Just as you left my quote there intact, no one (barring any > exception I don't recall) claimed to be giving a DEFINITION, No one NEEDS to be CLAIMING to be GIVING a definition IN ORDER to be RELYING on one! Just because you are not CLAIMING to be giving a definition does NOT mean "your definition is wrong -- here is the right one" is unAVAILABLE as a criticism! > and their > formulas (not including the original one that people were responding > to) were correct No, they weren't. > and you, as usual, made a bloody foaming fool of > yourself. In your ignorant opinion. Clue: quality is more important than quantity here. The relevant outcome is not going to be determined by how MANY people think I am being more foolish than you are. Knowledge is republican, not democratic. > > That ZFC does not have unrestricted > > comprehension does not contradict > > that the formulas posted by the poster > >G. Frege and some other people were correct > > It does so too. Then show a contradiction. > > - are theorems of ZFC - > > They INCLUDED TERMS which ZFC *does*NOT*construct*. > They were as such not even grammatical within the context of ZFC. Quite ordinary and commonly understood usage of set abstraction notation easily allows such terms. > > and were not claimed to be definitions. > > Which is of course utter bullshit -- EVERY time you write > a sentence (theorem or otherwise), EVERY closed term > in it is a definition of its referent. Bizarre notion in context of ordinary mathematical logic and set theory. > > > > But what makes it egregious is that he was, in his usually charming > > > > way, storming in tell everybody else that they're wrong (they > > > > weren't), and even breathing fire about getting the definition wrong > > > > when it wasn't even a DEFINITION that was at stake. > > The definition of the empty set WAS at stake in that thread. You only insist. > I am not going to belabor the point vs. anyone who chooses > to deny this; anybody whose respect I care to care about > will be able to see it for himself. Who cares who you respect? > > Right. Just as you left my quote there intact, no one (barring any > > exception I don't recall) claimed to be giving a DEFINITION, > > No one NEEDS to be CLAIMING to be GIVING a definition IN ORDER > to be RELYING on one! And none of the people such as the poster G. Frege (i.e., the people who came in to correct the original poster who was mixed up) were seen to be relying on an incorrect definition. > Just because you are not CLAIMING to be > giving a definition does NOT mean "your definition is wrong -- here is > the right one" is unAVAILABLE as a criticism! I didn't say such a thing is unavailable as a criticism. Meanwhile, the remarks by such people as G. Frege were correct. > > and their > > formulas (not including the original one that people were responding > > to) were correct > > No, they weren't. > > and you, as usual, made a bloody foaming fool of > > yourself. [quoted text clipped - 5 lines] > how MANY people think I am being more foolish than you are. > Knowledge is republican, not democratic. I never argued that correctness in the matter we are discussing now is determined by poll. MoeBlee > > > and were not claimed to be definitions. And it occurred to me, even if the formulas were offered as definitions (though they were not, at least not by the posters who corrected the original poster), such a formula as: 0 = {x | ~x=x} is found as a definition in certain widely referenced textbooks (as well as a theorem in many others). MoeBlee > such a formula as: > > 0 = {x | ~x=x} > > is found as a definition in certain widely referenced textbooks (as > well as a theorem in many others). We WERE NOT IN the context of those textbooks. Jeezus. We were talking about what could be proved/constructed. In the context of ZFC it is necessary to make the construction clear from the axioms, and pedagogically, it is equally necessary to use notation that does NOT obscure that clarity. Unrestricted comprehension was NOT allowed in the context that WE were in. I personally don't give a f.ck how many textbooks you think you can cite, since even when citing something as authoritative as Enderton, you can still choose to cite it for bad reasons and to bad ends. > > such a formula as: > [quoted text clipped - 5 lines] > We WERE NOT IN the context of those textbooks. > Jeezus. The point of mentioning them is that what the posters wrote was correct, though you claimed it incorrect, and that is seen by just looking at a few textbooks. > We were talking about what could be proved/constructed. > In the context of ZFC it is necessary to make the construction > clear from the axioms, and pedagogically, it is equally necessary > to use notation that does NOT obscure that clarity. The posters gave a result; they didn't claim also to give proofs. And that result is a theorem in some treatments and a defintion in others, so correct in any case. And not every post needs to be a SPOON FEEDING of utter pedagogical efficacy. The original poster was incorrect, and the followup posts correctly commented. > Unrestricted comprehension was NOT allowed in the context > that WE were in. And no one used it. > I personally don't give a f.ck how many textbooks > you think you can cite, since even when citing something as > authoritative as Enderton, you can still choose to cite it for > bad reasons and to bad ends. I've not chosen any bad reasons or bad ends. By the way, Enderton is one of those who mentions 0 = {x | ~x=x}. There's nothing wrong in saying 0 = {x | ~x=x}, and no bad reasons or bad ends for doing so, and no pedagogical harm in it. MoeBlee > I just let go by uncontested thousands of words of yours, some of them > HILARIOUSLY ill-conceived (I just love your 0 = {x | Ay ~yex}, which > you posted to CORRECT a bunch of people who had ALREADY CORRECTLY > observed various equations with 0 !!!) That was simply a mistake. Everybody (including you) excused it as such UNTIL NOW. You are lapsing. You had it right the first time when you just let it go by. And there was not any "already correctly" going on in that context. Nam had said one thing wrong and FF had (as usual) corrected it INcorrectly. > and also intellectually > hypocritical (the line you're arguing about semantics and axioms > lately is the EXACT NEGATION of the line you argued, rather by > spraying your mouth-foam in my face, when we first exchanged posts), Put up or shut up. I do not go around posting undocumented lies about people. Since you now appear to be doing exactly that, I will be content for us to rmain enemies. > since I learned a while ago that not only is there no point trying to > get through to you but doing so is an ESPECIALLY unpleasant endeavor. I certainly don't mind people who are more ignorant than I am "giving up" on trying to persuade me to agree with their errors. I do, however, mind people lying about me or claiming that I said X without being able to back it up. > Nam had said one thing wrong and FF had (as usual) > corrected it INcorrectly. > :-) Keep up the good worke, george. F. SignatureE-mail: info<at>simple-line<dot>de > > I just let go by uncontested thousands of words of yours, some of them > > HILARIOUSLY ill-conceived (I just love your 0 = {x | Ay ~yex}, which > > you posted to CORRECT a bunch of people who had ALREADY CORRECTLY > > observed various equations with 0 !!!) > > That was simply a mistake. Sure, we all make mistakes. What makes it notable though is the context of you harranging a bunch of other people who were not mistaken. > Everybody (including you) excused it as such UNTIL NOW. > You are lapsing. You had it right the first time when you just > let it go by. And there was not any "already correctly" going on in > that > context. Nam had said one thing wrong and FF had (as usual) > corrected it INcorrectly. Nam is out to lunch; we know that. But the poster Frege's remarks were correct. He didn't say his formulation is a DEFINITION of the empty set. > > and also intellectually > > hypocritical (the line you're arguing about semantics and axioms > > lately is the EXACT NEGATION of the line you argued, rather by > > spraying your mouth-foam in my face, when we first exchanged posts), > > Put up or shut up. The thread you renamed to 'tedious sledding [...]' and your crazed attack of my perfectly sensible distinctions regarding provability and semantics as opposed to your latest mood swings on that subject. > I do not go around posting undocumented lies about people. > Since you now appear to be doing exactly that, I will be content > for us to rmain enemies. Oh good, a toxic dump is declaring me an enemy. > > since I learned a while ago that not only is there no point trying to > > get through to you but doing so is an ESPECIALLY unpleasant endeavor. [quoted text clipped - 4 lines] > I do, however, mind people lying about me or claiming that I said > X without being able to back it up. You seriously deflated the meaning of 'lying' a long time ago. MoeBlee > > > and also intellectually > > > hypocritical (the line you're arguing about semantics and axioms [quoted text clipped - 6 lines] > attack of my perfectly sensible distinctions regarding provability and > semantics as opposed to your latest mood swings on that subject. This is still not putting up or shutting up, but obviously I have a preference between the two. > > > > and also intellectually > > > > hypocritical (the line you're arguing about semantics and axioms [quoted text clipped - 9 lines] > This is still not putting up or shutting up, > but obviously I have a preference between the two. Anyone interested, including you, can just read your sustained rants in that thread I mentioned compared with your sustained rants in the last couple of months on the subject. And, since sorting through your garbage is not something I enjoy doing, my preference is to let it stand there. MoeBlee > > > and also intellectually > > > hypocritical (the line you're arguing about semantics and axioms [quoted text clipped - 5 lines] > > The thread you renamed to 'tedious sledding [...]' Lying as usual. The argument in that first encounter was about the existence of "logical axioms", which I deprecated in favor of inference rules. I was making the point that the usual inference rules for the quantifier in 1st-order logic have semantic content. You were claiming that axioms don't have semantic content, but my position then does not contradict my position now: EVEN THEN, I was insisting that the things with semantic content were inference rules AND NOT axioms. > > > > and also intellectually > > > > hypocritical (the line you're arguing about semantics and axioms [quoted text clipped - 12 lines] > semantic content. You were claiming that axioms don't have > semantic content, I made no statement that I accept as summarized as "axioms have no semantic content". And your arguments then were not just about axioms and a preference for inference rules but also as to additional concerns regarding syntax and semantics, while, by the way, you were strawmaning me in prosecution of your arguments against things I wasn't even claiming, AS YOU SO USUALLY DO. > but my position then does not contradict > my position now: Like I said, your garbage is best left unsorted. > EVEN THEN, I was insisting that the things > with semantic content were inference rules AND NOT axioms. Bizarre. MoeBlee > Like I said, your garbage is best left unsorted. You can't just point at a garbage-heap and allege that there is something in it that proves I was an a.shole. Well, YOU can, but hopefully, that's just you. > > > Okay, in a technical sense, '2=1+1' is true relative to models because > > > it's true in some models but not in others. [quoted text clipped - 6 lines] > too distant past. The phrase 'true in a model' is routine and vastly > understood. But not by Nam. It matters. > And though less routine, upon just a bit of explanation, it can be > seen that it turns out (it is a theorem) that every structure for a > language is a model (of some theory) and, of course, by definition, > every model of a theory is a structure (for some language). But that is completely trivial and irrelevant. The point that Nam NEEDED made to him and that you did NOT make was that "2=1+1" is NOT "true in some models" OF PA "but false in others". "2=1+1" is true in ALL models of PA because 2=1+1 is a THEOREM of PA. There are not ANY *relevant* "models" in which 2=1+1 is false. THIS IS IMPORTANT. You did NOT get this right in arguing with Nam, or with me. You just spent a few more paragraphs defending bullshit and attacking my character. Thus, > theorem: M is a model iff M is a structure. A model by definition NEEDS to be a model OF something. TO USE "Model" so generically as to imply that it means the same thing as "structure" IS f.cking STUPID. > This is derivable from > defintions of these terms as given in Enderton's 'A Mathematical > Introduction To Logic', though perhaps he does not make a point of > mentioning it. OF COURSE NOT, BECAUSE IT IS f.cking STUPID. The only reason YOU are making a point of it NOW is TO TRY TO WIN A PISSING CONTEST. Just go to hell. The point THAT MATTERS is that NOBODY SHOULD EVER refer to "models" generically. THAT'S PRECISELY WHAT "structures" and "interpretations" are FOR. You don't SWITCH from either of those TO models UNTIL you are READY AND WILLING to say WHAT your model is a model OF! Quoting Enderton to try to justify the opposite of this is intellectual hypocrisy on YOUR part, if you want to keep accusing people of that. Your behavior here (from a debator/discussant standpoint) has just been utterly deplorable. > > > > Okay, in a technical sense, '2=1+1' is true relative to models because > > > > it's true in some models but not in others. [quoted text clipped - 8 lines] > > But not by Nam. It matters. Nam misunderstands a lot of things. If you look at the CONTINUATION of my remarks you will see how they guide him back to another important aspect of this matter, though I don't hold any special hope that he will ever untangle himself from his confusions. > > And though less routine, upon just a bit of explanation, it can be > > seen that it turns out (it is a theorem) that every structure for a > > language is a model (of some theory) and, of course, by definition, > > every model of a theory is a structure (for some language). > > But that is completely trivial and irrelevant. It's trivial. Which is added to MY point, which is just to say that I used the terminology properly. > The point that Nam > NEEDED made to him and that you did NOT make I made a certain point. My purpose in posting is not always to make the exact certain point that you think is needed to make or that is even the very most important point to make in the context. Anyway, if you look at the CONTINUATION of my remarks you'll see how they would guide Nam back to a better understanding were he disposed to making an attempt to think about this subject clearly. My remarks pertained to a certain aspect of the matter under discussion; it is not required that all remarks go directly to the exact points that you consider to be of greatest importance. > was that > "2=1+1" is NOT "true in some models" OF PA "but false in others". > "2=1+1" is true in ALL models of PA because 2=1+1 is a THEOREM > of PA. There are not ANY *relevant* "models" in which 2=1+1 is false. > THIS IS IMPORTANT. Aside from the notion of what a 'relevent' model is, nothing I said contradicts that '2=1+1' is true in all models of PA or would lead to thinking otherwise or even distract from that fact. > You did NOT get this right in arguing with Nam, or with me. > You just spent a few more paragraphs defending bullshit Nothing I posted is incorrect to properly be described as 'bullshit' > and > attacking my character. Your character is obnoxious and ludicrous to me. From time to time I like to mention certain particulars in that regard. > > theorem: M is a model iff M is a structure. > > A model by definition NEEDS to be a model OF something. > TO USE "Model" so generically as to imply that it means the > same thing as "structure" IS f.cking STUPID. M is a model iff M is a structure such that there is a set of sentences all of which are true in M. There is no harm in saying "M is a model" without specifying which sets of sentences M is a model of. > > This is derivable from > > defintions of these terms as given in Enderton's 'A Mathematical [quoted text clipped - 4 lines] > The only reason YOU are making a point of it NOW is > TO TRY TO WIN A PISSING CONTEST. As to pissing contests, I already told you that I concede the power of your bladder, indeed your ability to inundate vast areas, let alone the power of your spleen. And I mentioned the terminological point just to show that my use is sensible as it came up in other remarks I was making. > Just go to hell. Reading your posts is hell enough. > The point THAT MATTERS > is that NOBODY SHOULD EVER refer to "models" [quoted text clipped - 4 lines] > opposite of this is intellectual hypocrisy on YOUR part, if you want > to keep accusing people of that. Your continual SHOULDS are as noise and litter to me. Even in Chang & Keisler will you find things along the lines of 'M is a model for the language L' as well as 'M is a model of the theory T'. And it was my very point that 'M is a model for the language L' is acceptable as 'M is a structure for the language L'. > Your behavior here (from a debator/discussant standpoint) has just > been utterly deplorable. Will that comment go on my permanent record? I shudder to think... MoeBlee >> tc...@lsa.umich.edu wrote: >>>> tc...@lsa.umich.edu wrote: [quoted text clipped - 23 lines] > > (1) No, you completely dodged the point. No I'm not. See my latest response to TC. > Okay, in a technical sense, '2=1+1' is true relative to models because > it's true in some models but not in others. In a technical sense, "2=1+1" is true in all models of some consistent theories where it is a theorem. > But the challenge put to you was to say in what way (3) is relative to > models, since it's not a matter of being true in some models and not [quoted text clipped - 5 lines] > (2) What is the relativity in '0011' being the same string as '0022 > with 1 substituted for 2'? Quite a few ways of relativity! For example, what did *you* mean by "substituted"? Were you referring to some kind of _mapping_? > MoeBlee > In a technical sense, "2=1+1" is true in all models of some consistent > theories where it is a theorem. I might sound strange, but let me re-phrase it: In a technical sense, "2=1+1", is *supposed to be* true in all models of some consistent theories where it is a theorem. (I know there exists a meta theory called "Completeness"). > >> tc...@lsa.umich.edu wrote: > >>>> tc...@lsa.umich.edu wrote: [quoted text clipped - 25 lines] > > No I'm not. See my latest response to TC. There you just spin more variations on your basic misunderstandings. > > Okay, in a technical sense, '2=1+1' is true relative to models because > > it's true in some models but not in others. > > In a technical sense, "2=1+1" is true in all models of some consistent theories > where it is a theorem. What you just said boils down to: '2=1+1' is true in all models of '2=1+1'. Yeah, we know that. > > But the challenge put to you was to say in what way (3) is relative to > > models, since it's not a matter of being true in some models and not > > in others, but rather of being true PERIOD, since it is a statement > > about ALL models. To reinforce that it is a statement about all > > models, please recognize that it is of the form, Given ANY model, if > > it is a model of PA, then [...]. You didn't answer that. > > (2) What is the relativity in '0011' being the same string as '0022 > > with 1 substituted for 2'? > > Quite a few ways of relativity! For example, what did *you* mean by "substituted"? > Were you referring to some kind of _mapping_? I mean it in the ordinary English sense of the word. If you point is that words can be defined differently, so that the truth of a statement is relative to how we define the words, then I don't think anyone would disagree with you. 'Meryl Streep has blonde hair' is true depending on what we mean by 'Meryl Streep', 'has', 'blonde' and 'hair'. But I don't think that is what is at issue. Given the ordinary informal English meanings of the words used in (2), don't you think it is a non-religious, objective observation that they are the same string? MoeBlee <tchow@lsa.umich.edu wrote: <> (3) In every model of PA, there are no nonzero integers m and n such that <> m^2 = 2 n^2. [...] <Apparently you've missed my last post where I made the correction: < < > Let me re-phrase it: what might be "integers" or "standard" to one, < > might not be to the others. < <Any rate, so (3) is true, *relative* to PA's models: the *truth* of (3) <is a relative value, and the relativity is still there! What do you mean that (3) is true relative to PA's models? What else can (3) be relative to? Or to ask the question another way, explain to us a sense in which (3) can be false. SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences > <tchow@lsa.umich.edu wrote: > <> (3) In every model of PA, there are no nonzero integers m and n such that [quoted text clipped - 9 lines] > > What do you mean that (3) is true relative to PA's models? I might just have been too quick here. Yes my statement "(3) is true, *relative* to PA's models" sounds a bit obscure in meaning. So let me explain it. But first, iirc, the context of our debate is you (and others) believe certain mathematical truths, and just like religion beliefs, are *absolutely* true that we could not believe or see otherwise. On the other hand I believe that in so far as any mathematical statement has to be stated/worded (whether it's a FOL formula or a meta statement), there's *nothing absolute* about it's truth: there is always hierarchy of framework contexts that you could traverse upward one level and go down in the other direction to see the opposite truth *for the very same stated statement*! (But religion statement is not supposed to be as such!). In other words, there is *no intrinsic (built-in) semantic for a sentence* (FOL formula, meta statement, or any sentence for that matter). Its semantic and hence truth is always relative to some context! > What else can (3) be relative to? Given what I've just stated above, then there are more than one way the truth of (3), as a meta statement, could be altered. For an example, if we simply change some of the reasoning framework contexts, such as rules of inference, the truth of (3) in principle could be changed. > Or to ask the question another way, explain to us a sense in which (3) can be false. For another example, if we change by what we mean by "PA" then (3) is not necessarily true. So, sure, one could accuse me of talking what would typically considered as "frivolous" here. On the other hand, talking about mathematical "absolute" beliefs/truths as if they were religious ones then I think everything is a fair game! >> <> (3) In every model of PA, there are no nonzero integers m and n such that >> <> m^2 = 2 n^2. [...] >For another example, if we change by what we mean by "PA" then (3) is >not necessarily true. But then what kind of distinction is there between mathematics and religion? Consider the following religious statement: (4) Jesus is the son of God. By changing the meaning of "Jesus," "son," and "God," the truth value of (4) could change. Maybe I have a cat named "God" and one of its litter is named "Jesus." Or maybe I have a dog named "God" with no offspring. If this is what you mean by relativity, then the technicalities of mathematical logic are irrelevant, and there is no distinction between mathematical statements and religious statements in this regard. SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences >>> <> (3) In every model of PA, there are no nonzero integers m and n such that >>> <> m^2 = 2 n^2. [quoted text clipped - 10 lines] > could change. Maybe I have a cat named "God" and one of its litter is named > "Jesus." Or maybe I have a dog named "God" with no offspring. The distinction is, in religion, a statement like (4) has only *one semantic* and is of *one truth*. Wars have been waged because someone named a dog "God" or there is a belief that (4) is not true, so to speak. But it's perfectly normal for us to speak of, say, "1+1=0" as true, or false, relative to whatever the context that we choose. (But there's *no absolute context* that everyone must accept that statement as true or otherwise). > If this is what you mean by relativity, then the technicalities of mathematical > logic are irrelevant, and there is no distinction between mathematical statements > and religious statements in this regard. I'd respectfully think this is a mis-characterization of "the technicalities of mathematical logic [reasoning]", on at least 2 counts: the *history* and the *essence* of mathematical reasoning. On the account of history, there were times in the past we believed the 5th postulate be "absolutely" true or that there being no "number" whose square is 1 be "absolutely" not false. The point is history seems to have shown us the value of mathematical reasoning should be with consistency of arguments, rather than with the truth of what's formally uttered or stated. For the notion of truth is always a relative notion: any statement that's considered as true in some way would always be false, in some other (legitimate) ways! On the account of essence, mathematical reasoning is based on *finite knowledge*: from finite formula length to finite proofs. Given such finite foundation of reasoning, would you think in this case we know enough about a particular *infinite* set of axioms so that when 2 persons refer to "PA", they could be certain they are referring to 2 identical sets of axioms? If not, then isn't it true that meaning of "PA" would be relative to what exact axiom set they *each* might think they know in their individual mind? And so in this context, any statement about PA is a relative one, relative to the exact individual "PA"'s axiom-set that happens in one's mind. And this relativity is a legitimate one. Unlike that, to be a *religion statement*, (4) is supposed to have one meaning and one truth that is universally acknowledged and that's immutable, *in all contexts*. > On the account of history, there were times in the past we believed the > 5th postulate be "absolutely" true or that there being no "number" whose > square is 1 be "absolutely" not false. Sorry for a typo. I meant "whose square is -1". "Nam D. Nguyen" <namducnguyen@shaw.ca> ha scritto >>>> <> (3) In every model of PA, there are no nonzero integers m and n such >>>> that [quoted text clipped - 18 lines] > semantic* > and is of *one truth*. It depends on the language we are using to speak about religion. If we are speaking spanish (4) has no meaning. If we are speaking "Gunglish", a new language that is equal to english for everything but the word "son", that means "father", then (4) is false. So the truth value of (4) is still relative. To fix one meaning you have to fix the language, but this would also fix the meaning of "PA" and mathematics wouldn't also be relative anymore. > "Nam D. Nguyen" <namducnguyen@shaw.ca> ha scritto > [quoted text clipped - 29 lines] > To fix one meaning you have to fix the language, but this would also fix > the meaning of "PA" and mathematics wouldn't also be relative anymore. Consider: (4') Jesus ist der [einzige] Sohn Gottes. (4) and (4)' are supposed to have the identical *religious* "meaning" and "truth", which are beyond isomorphism of wordings and languages, so to speak. Otoh, "1+1=0" and "1' +' 1' = 0'" not only could differ in meaning and truth per languages, but also could differ in the same language. E.g., "1+1=0" could differ between the natural "arithmetic" and the modulo counterpart!. (And that's the difference between absolute meaning of religous truths and mathematical truths, imho). >But it's perfectly normal >for us to speak of, say, "1+1=0" as true, or false, relative to whatever the >context that we choose. (But there's *no absolute context* that everyone must >accept that statement as true or otherwise). It still seems to me that your view suffers from a problem of infinite regress. You don't think that statements such as "1+1=0" or "PA is consistent" have a determinate truth value; one must fix the context. However, what if we fix the context? *Then* do we get a determinate truth value? So for example, consider (*) "1+1=0" is false in the standard model of PA. Does (*) have a determinate truth value? I've fixed the context of "1+1=0", haven't I? Doesn't this thereby fix the truth value of (*)? Doesn't it make the "belief having been believed already" (or whatever locution it was you used)? But when I pressed this point before, you said no; "PA," you said, was itself indeterminate. So somehow we have to fix the context of *which* PA we're talking about. But now an infinite regress looms. Won't any attempt to fix the context of PA itself use concepts whose context needs fixing? Does the process ever bottom out? If not, then it never makes sense to say something is true or false, *even relative to a context*, because there's no way to specify a context determinately enough to fix a truth value for the original sentence. If on the other hand the process *does* bottom out, then doesn't the "bottomed-out" statement have a determinate truth value in an absolute sense? (There are many choices of context, and perhaps there is no privileged choice, but once a context is fixed, the truth value of the fixed-context statement is *absolute*.) SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences >> But it's perfectly normal >> for us to speak of, say, "1+1=0" as true, or false, relative to whatever the [quoted text clipped - 3 lines] > It still seems to me that your view suffers from a problem of infinite > regress. The meaning of relativity is that *we* (all finite beings) would have truth-views that suffer from this infinite regress, and there isn't much we could do about changing it. > You don't think that statements such as "1+1=0" or "PA is > consistent" have a determinate truth value; one must fix the context. > However, what if we fix the context? *Then* do we get a determinate > truth value? I'd think "No". If we've relativized a truth to a context then we'd have a relativized truth, not an absolute (i.e. "determinate") one. > So for example, consider > > (*) "1+1=0" is false in the standard model of PA. > > Does (*) have a determinate truth value? I've fixed the context of > "1+1=0", haven't I? Have you [a general "you"] precisely spelled out *all* the axioms of "PA"? If not, then you haven't fixed a precise context for (*). > Doesn't this thereby fix the truth value of (*)? And suppose we did manage to fix the context of (*), we've just fixed a *relative truth*, relative to the context of exactly which "PA" we've chosen. > Doesn't it make the "belief having been believed already" (or whatever > locution it was you used)? But when I pressed this point before, you > said no; "PA," you said, was itself indeterminate. So somehow we have > to fix the context of *which* PA we're talking about. That phrase "belief having been believed already" refers to a relative truth, one that has been already "relativised". The point I'm trying to make is that there is no absolute truth-value for (*), because, among other things, *we* *can not spell out all* the axioms: each of us might think of different set of axioms that could be named as "PA" set! And the moment each of us thinks "PA"-set has been fixed, the moment that truth value of (*) has been a relative one! > But now an infinite regress looms. May I say "Welcome to the relativity nature of (human) mathematical reasoning"! > Won't any attempt to fix the context of PA itself use concepts whose context > needs fixing? Does the process ever bottom out? "Yes" to the 1st question; and "No" to the 2nd one. I might add though there are 2 kinds of contexts where relativity of mathematical truths (in general) could take place: inheritance (parental) context, and lateral (sibling) context. Whether or not it's FOL= we're choosing is the first kind of context, while whether or not (within say FOL=) we agree which *specific* axiom-set to be referred as "PA" is the 2nd kind of context. Absolutism in mathematical truth values is impossible because both the the semantic and truth of a statement (FOL or meta) could in general be altered in either one of these 2 contexts (or even in a combination of both!). > If not, then it never makes sense to say something is > true or false, *even relative to a context*, because there's no way to > specify a context determinately enough to fix a truth value for the > original sentence. Agree. Only if we *insist* on "absolute sense" (whatever that might or might not mean), would we have to be agonized on what anything would "absolutely" make sense or be true. If, otoh, we contend that meaning and truth of any grouping of symbols is relative, we might go a long way in using mathematical language as a descriptive tool, describing what we'd (relatively) perceive as "the" reality, imho. > If on the other hand the process *does* bottom out, > then doesn't the "bottomed-out" statement have a determinate truth value > in an absolute sense? (There are many choices of context, and perhaps > there is no privileged choice, but once a context is fixed, the truth > value of the fixed-context statement is *absolute*.) As mentioned, that would be a "fixed" relativized truth. >Have you [a general "you"] precisely spelled out *all* the axioms of "PA"? >If not, then you haven't fixed a precise context for (*). PA isn't finitely axiomatizable; is that important to you? If so, let's use NBG instead of PA. NBG is finitely axiomatizable, and so we can indeed spell out *all* the axioms of NBG. >And suppose we did manage to fix the context of (*), we've just fixed >a *relative truth*, relative to the context of exactly which "PA" we've >chosen. Fine. So do you agree that there exist relativized truths? For example: (***) In every model of NBG, there do not exist nonzero integers a and b such that a^2 = 2 b^2. Because I'm using NBG, there is no problem with spelling out all its axioms explicitly. Do you agree that (***) is a relativized truth? If so, then do you *believe* (***)? If so, then on what basis do you believe (***)? Most people would say that it's because of the *proof* that sqrt(2) is irrational. But not you, I suppose? SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences >> Have you [a general "you"] precisely spelled out *all* the axioms of "PA"? >> If not, then you haven't fixed a precise context for (*). > > PA isn't finitely axiomatizable; is that important to you? As long as there is a distinct possibility ~GC isn't provable in Q, then "Yes" it seems important to me. > If so, let's use > NBG instead of PA. NBG is finitely axiomatizable, and so we can indeed spell > out *all* the axioms of NBG. Or Q. I don't know about every intelligent beings in the world (or universe) but relatively to me I could spell out all axioms of Q. And I suspect that relatively to you, a complete spell-out of NBG/Q can be done easily. So relatively speaking, the complete spell-out here has a context it seems. >> And suppose we did manage to fix the context of (*), we've just fixed >> a *relative truth*, relative to the context of exactly which "PA" we've [quoted text clipped - 9 lines] > > If so, then do you *believe* (***)? I don't much about *all* beings there could exist, I do believe in (***) as a relative truth, relatively to me as an being who could do some mathematical reasoning and who's *assuming* certain contexts, e.g. FOL= rules of inference, etc... > If so, then on what basis do you believe (***)? Most people would say that > it's because of the *proof* that sqrt(2) is irrational. But not you, I > suppose? Oh it's because of proof too, relative to my reasoning. But of course *proof* is always relative to rules of inference, among other things. And there simply are *no "absolute" rules of inference*! >> (***) In every model of NBG, there do not exist nonzero integers a and b >> such that a^2 = 2 b^2. [...] >I don't much about *all* beings there could exist, I do believe in (***) >as a relative truth, relatively to me as an being who could do some >mathematical reasoning and who's *assuming* certain contexts, e.g. >FOL= rules of inference, etc... Good. Now we can return to the point where I entered the discussion, when you said, `Unfortunately mathematical reasoning isn't religion where "beliefs" would be much relevant.' But you state here plainly that you believe in (***) as a relative truth, relative to blah blah blah. Therefore "belief," properly understood and qualified, *is* relevant to mathematics, contrary to what you asserted previously. I rest my case. SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences >>> (***) In every model of NBG, there do not exist nonzero integers a and b >>> such that a^2 = 2 b^2. [quoted text clipped - 10 lines] > understood and qualified, *is* relevant to mathematics, contrary to what you > asserted previously. What I've asserted from the beginning and through out is *religious belief* is supposed to be absolute and is not the relative kind of beliefs we employ in mathematical reasoning. That's all I've meant to say! In any rate, since truth and no truth could be equated to consistency and inconsistency, respectively, of syntactical axioms, truth and no truth is *relative* to *which set* the axioms are in! > I rest my case. I might open the case again if anyone believes there is absolute truth in reasoning! >What I've asserted from the beginning and through out is *religious belief* is >supposed to be absolute and is not the relative kind of beliefs we employ in >mathematical reasoning. That's all I've meant to say! Perhaps that's all you *meant* to say, but what you *actually* said was: Unfortunately mathematical reasoning isn't religion where "beliefs" would be much relevant. in response to Daryl McCullough's comment: So, for example, a proof in PA + the negation of Goldbach's conjecture would not be very convincing, because we have no reason to believe that the negation of Goldbach's conjecture is true. The "belief" in question, understood as a relative belief in a relative truth relative to blah blah blah, is entirely relevant. SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences >> What I've asserted from the beginning and through out is *religious belief* is >> supposed to be absolute and is not the relative kind of beliefs we employ in [quoted text clipped - 13 lines] > The "belief" in question, understood as a relative belief in a relative truth > relative to blah blah blah, is entirely relevant. At this point in time, when there's no proof, the belief that the negation of Goldbach's conjecture is not true is of the nature of religion belief, which is quite irrelevant to reasoning: reasoning requires proof not belief. Wouldn't you think so? >>> What I've asserted from the beginning and through out is *religious >>> belief* is [quoted text clipped - 20 lines] > religion belief, which is quite irrelevant to reasoning: reasoning > requires proof not belief. Wouldn't you think so? In fact, DMC's whole statement above is a religion statement, dressed in formalism! A proof is supposed to be the only reason for belief. One could disprove a proof by proving the negation and proving the theory is consistent. But one should *not* dismiss a proof simply because one already has the opposite belief, *without* a proof! >>> in response to Daryl McCullough's comment: >>> >>> So, for example, a proof in PA + the negation of Goldbach's conjecture >>> would not be very convincing, because we have no reason to believe that >>> the negation of Goldbach's conjecture is true. [...] >> At this point in time, when there's no proof, the belief that the >> negation of Goldbach's conjecture is not true is of the nature of [quoted text clipped - 6 lines] >theory is consistent. But one should *not* dismiss a proof simply because >one already has the opposite belief, *without* a proof! Your remarks here are even more absurd than your remark that belief is not relevant. Daryl McCullough is not asserting that the negation of Goldbach's conjecture is not true He asserts only that we have no reason to believe that the negation of Goldbach's conjecture is true That is, he is withholding belief in the negation of Goldbach's conjecture, because there is no proof of that statement. In other words, he is behaving in exactly the way that you say he should. So what are you complaining about? Obviously you're just not reading what he wrote very carefully. SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences >>>> in response to Daryl McCullough's comment: >>>> [quoted text clipped - 21 lines] > we have no reason to believe that the negation of Goldbach's conjecture > is true I'm giving you one clue: had he only said exactly no more than that sentence, then what you've said would have been true; and in fact I would have not involved in this conversation at all! He did use the words "_for example_", "_because_" didn't he? The point being is he was making an (meta level) *assertion*/*argument* with a hypothesis that would lead to a conclusion, however informal his whole statement may have sounded. And since his " we have no reason to believe... is true" is a *supporting hypothesis*, one would have no choice but interpreting it as the assertion "the negation of Goldbach's conjecture is not true". > That is, he is withholding belief in the negation of Goldbach's conjecture, > because there is no proof of that statement. In other words, he is behaving > in exactly the way that you say he should. So what are you complaining about? One of the things I'd complain about his whole statement is the absurdity that *any* proof of ~GC in PA would be not convincing, without seeing that proof! As I mentioned, proof is very fundamental to what we should or should not believe. So how could one possibly say that as far as reasoning is concerned? And you're defending this "absurdity"? (It'd be quite a surprise to me!) > Obviously you're just not reading what he wrote very carefully. I think it's you, not I, who didn't read what he asserted or what has been being argued carefully. >And since his " we have no reason to believe... >is true" is a *supporting hypothesis*, one would have no choice but >interpreting it as the assertion "the negation of Goldbach's conjecture >is not true". Well, if you refuse to take what he said at face value, but insist on twisting his words into some totally different statement, then there's no arguing with you. I'm satisfied that you admit that you were totally wrong, *provided* we assume that Daryl meant what he said, and that you are correct only if we forcibly misinterpret Daryl as saying something completely different from what he actually said. SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences >> And since his " we have no reason to believe... >> is true" is a *supporting hypothesis*, one would have no choice but [quoted text clipped - 6 lines] > > I'm satisfied that you admit that you were totally wrong, You're bordering being dishonest. Where did I admit that? > *provided* we > assume that Daryl meant what he said, and that you are correct only if > we forcibly misinterpret Daryl as saying something completely different > from what he actually said. >>> And since his " we have no reason to believe... >>> is true" is a *supporting hypothesis*, one would have no choice but [quoted text clipped - 13 lines] >> we forcibly misinterpret Daryl as saying something completely different >> from what he actually said. What Daryl stated: "So, for example, a proof in PA + the negation of Goldbach's conjecture would not be very convincing, because we have no reason to believe that the negation of Goldbach's conjecture is true." So his hypothesis is: H = "we have no reason to believe that the negation of Goldbach's conjecture is true." His (reasoning) conclusion based on H is: C = "a proof in PA + the negation of Goldbach's conjecture would not be very convincing". Independent of what he actually intended to say, taken on face value his statement's conclusion is, after being stripped from its informality: C' = "(PA + ~GC) is an inconsistent theory due to ~GC" C' is very technical, H is not. So, if C' is "proven" on the basis a non-technical H, then C' is a religion-like conclusion: it has no reasoning basis. If H needs to be translated to a technical assertion (to be a hypothesis), how could it possibly be not equivalent to this following technical statement: H' = "the negation of Goldbach's conjecture is not true" or equivalently: H'' = "~GC is not provable from Q" ? But if so, H' or H'' would be a religion-like statement because there is no proof to it. So overall, the conclusion C or C' has no proof, no basis at all. Why TC didn't see it and reacted the way he did is really surprising to me. > What Daryl stated: > [quoted text clipped - 16 lines] > > C' = "(PA + ~GC) is an inconsistent theory due to ~GC" It is utterly incomprehensible that you read C' into what Daryl said. In any case not-GC is Sigma_1, PA is Sigma_1 complete, so if not-GC, then PA proves it and (assuming PA is consistent) so is PA + ~GC. If Daryl DID hold not-GC (and how you get from his saying that there is no reason to believe non-GC to his assuming it is a mystery), he -- being a sensible chap -- would be *denying* that (PA + ~GC) is inconsistent. But perhaps that's a typo for C" = "(PA + ~GC) is an inconsistent theory to GC But again Daryl, being a sensible chap doesn't say that either. He at most says that, if GC were true (and we have no reason to deny it), the theory PA + ~GC wouldn't be reliably truth-generating. But that OF COURSE doesn't entail that the theory is inconsistent. >> What Daryl stated: >> [quoted text clipped - 18 lines] > > It is utterly incomprehensible that you read C' into what Daryl said. "a proof" is "a general proof" which is equivalent to *any proof*. If I say to you "*Any proof* of a formal system T *would not be convincing*", and if, on the merit of the statement alone - *not* what I intended to say but *didn't clarify*, that doesn't mean T is not inconsistent, then what on Earth would that could possibly mean to *you*, logically speaking? (Remember during my conversation with TC about his own statement and up to my post above, Daryl didn't respond to what I said I all. Consequently (and I already mentioned it) I had no choice but took his statement for what it was stated, with all the "cleansing" [of informality] that one would typically do in a technical argument.) So it's not incomprehensible at all (let alone "utterly incomprehensible") that you could read C' into C, on the face value of it! Right? > In any case not-GC is Sigma_1, PA is Sigma_1 complete, so if not-GC, > then PA proves it and (assuming PA is consistent) so is PA + ~GC. If [quoted text clipped - 8 lines] > > But again Daryl, being a sensible chap doesn't say that either. Sorry, without him clarifying anything - at the time all of this was debated, I couldn't possibly know what was in his mind; and what he said is what he stated. It's that simple! > He at > most says that, if GC were true (and we have no reason to deny it), > the theory PA + ~GC wouldn't be reliably truth-generating. But that OF > COURSE doesn't entail that the theory is inconsistent. Suppose for a moment I incorporated his newly revealed intention into C, that *still doesn't invalidate the point I was arguing with TC*, which is basically that Daryl's overall statement is a religion [-like] statement simply because we don't know *for certain* Q has any model: it's a mere defaulted *assumption* (i.e. *belief*) that Q/PA has a model. And a religion-like statement doesn't mean it must necessarily have any thing to do with God (though I think it's said Godel once tried to prove Him exist, mathematically!), all that means is when you believe a statement is true/false but there is no concrete supporting proof from the framework you're using to reason. > It is utterly incomprehensible that you read C' into > what Daryl said. It's not *utterly* incomprehensible. It may be incomprehensible for most people, but the person who committed this particular mis-comprehension was Nam. If you find THAT comprehensible then quite a few other previously inexplicable things become understandable. Nam D. Nguyen says... >So his hypothesis is: > [quoted text clipped - 10 lines] > >C' = "(PA + ~GC) is an inconsistent theory due to ~GC" No, I never suggested that C' is an inconsistent theory. I said that it is possibly an unsound theory. That is, it can prove false claims. Unsound does not imply inconsistent, as PA + ~Con(PA) shows. -- Daryl McCullough Ithaca, NY > Nam D. Nguyen says... > [quoted text clipped - 16 lines] > that it is possibly an unsound theory. That is, it can prove false > claims. Unsound does not imply inconsistent, as PA + ~Con(PA) shows. As I've explained recently to PS, what you've now said (or clarified) is not what you stated before. You've now stated a different statement! > -- > Daryl McCullough > Ithaca, NY Nam D. Nguyen says... >> Your remarks here are even more absurd than your remark that belief is not >> relevant. Daryl McCullough is not asserting that [quoted text clipped - 5 lines] >> we have no reason to believe that the negation of Goldbach's conjecture >> is true Well, Tim is exactly right. I am no asserting that the negation of Goldbach's conjecture is not true, I am asserting that we have no reason to believe that the negation of Goldbach's conjecture is true. >I'm giving you one clue: had he only said exactly no more than that sentence, >then what you've said would have been true; What he said is true. >and in fact I would have not involved in this conversation at all! And we'd all be happier. >He did use the words "_for example_", "_because_" >didn't he? Yes, the negation of Goldbach's conjecture is an example of a statement that we have no reason to believe. >The point being is he was making an (meta level) *assertion*/*argument* >with a hypothesis that would lead to a conclusion, however informal his whole >statement may have sounded. And since his " we have no reason to believe... >is true" is a *supporting hypothesis*, one would have no choice but >interpreting it as the assertion "the negation of Goldbach's conjecture >is not true". I don't know why you say that you have no choice but to interpret it that way. Nobody made you interpret it that way, and that interpretation is wrong. >One of the things I'd complain about his whole statement is the >absurdity that *any* proof of ~GC in PA would be not convincing, I didn't suggest any such thing. You are being ridiculous. There *is* no proof of the negation of GC (as far as I know). That's why I said that there is no reason to believe the negation of GC. -- Daryl McCullough Ithaca, NY > Nam D. Nguyen says... > >> and in fact I would have not involved in this conversation at all! > > And we'd all be happier. Somewhere I think I've heard: to kill a dog we would just label it a "mad" dog! >> Nam D. Nguyen says... >> [quoted text clipped - 4 lines] > Somewhere I think I've heard: to kill a dog we would just label it a > "mad" dog! Seriously. In the thread "About Consistency in 1st Order Theories." there was this comment about what I posted then from DCU (Dec. 22nd, 2005): <quote> If you have a specific replacement for FOL in mind then you might get people to comment on it. But you don't, you're just sort of hunting around for the replacement that's going to make you happy. If you want people to help you with this you might start by trying to convince people that there's a _need_ for this radical new version of logic. Exactly what the objective is is not clear to me. It seems possible that you might want to call it something other than "logic". Because whatever it is, it seems that in the thing you're looking for the "logic" is going to vary from person to person, and I suspect it's going to seem to a lot of people like the whole point to _logic_ is to study _correct_ reasoning, which will _not_ vary from person to person. Now, the class of mathematical facts that a given individual is actually able to prove certainly varies from person to person. If you want to study that somehow fine, but that seems more a topic in something like psychology than pure logic. If I'm correct in thinking that in the system you have in mind the _definition_ of correct reasoning is going to vary from person to person that seems even less like "logic". </quote> For 2 years I've tried in various posts/threads to follow his suggestion: - I've pointed out the relativity nature of current FOL reasoning is upon us all. I'd be for us all - not just me alone - to see it as it is so *we'd all* feel happy to make appropriate adjustment to this obsolete post Godel reasoning foundation. - I've pointed out that there can't be global "_correct_ reasoning" that *all reasoning beings could possibly know! - I've pointed out some suggestion how a new framework could be achieved: in a nut-shell: *) through formally recognizing certain limitation of "knowing" in some new (suggested) un-knownability principles. [One could see the section "A working perspective" from http://en.wikipedia.org/wiki/Foundational_crisis_of_mathematics for certain similarity of the "un-knownability principles" I've alluded to above. - I've similarly discussed many issues related to the above. But to be honest, for the past 2 years, many times I feel like I happened to enter a time machine going to Euclid's time where all the "professionals" would only care to utter one "intrinsic" truth: "The 5th postulate is 'absolutely' true'"! Of all the "science" fields, mathematical reasoning is supposed to be a field where we should keep up the motto: keep an open mind! It's kind of sad here in this forum most of the time we'd rather like to engage in a "fight" than in an open-mined scrutiny on mistakes of the past! Nam D. Nguyen says... >In fact, DMC's whole statement above is a religion statement, dressed in >formalism! You are a very strange person. -- Daryl McCullough Ithaca, NY Nam D. Nguyen says... >At this point in time, when there's no proof, the belief that the negation of >Goldbach's conjecture is not true is of the nature of religion belief, which is >quite irrelevant to reasoning: reasoning requires proof not belief. Wouldn't you >think so? I think that you are very confused. Belief in general has nothing to do religion. Religion is a particular *kind* of belief. So you are mixed up about that. When I agree to fly in an airplane, I am tacitly proclaiming that I believe that it is reasonably safe to do so. That has nothing to do with religion. When you claim "Reasoning requires proof not belief" you are espousing a *belief*. It's not a religious belief, but it is a belief. But also, you don't understand reasoning very well, either. Reasoning is certainly not the same thing as theorem proving. Outside of the limited domain of pure mathematics, reasoning is ultimately about trying to make the best decisions. Do I take this drug to prevent that disease? Is it safe to ride in this car or that airplane? What is the best shape for a bridge? You can work out the consequences of this or that theory, or of this or that assumption, but ultimately, when you make a decision, there will be unproved assumptions that will go into your decision. Refusing to make assumptions without proof is not reasoning, it is irrationality. We can weed out some assumptions as false because they are contradictory. Other assumptions we can prove are true. But at some point, you have to make a decision that depends on hypotheses that are not proven to be true. You can weigh the evidence for and against such a hypothesis, but conclusions are always uncertain. In the case of Goldbach's Conjecture, nothing much rides on its truth or falsity (as far as I know). It's not *necessary* to have any opinion either way about its truth value. However, it doesn't *hurt* anything, either. You are allowed to have a belief about anything. It's very bizarre for you to have such a strong reaction to my beliefs. I don't even have a very strong belief about -- Daryl McCullough Ithaca, NY > (***) In every model of NBG, there do not exist nonzero integers a and b > such that a^2 = 2 b^2. If a/b is a fraction in lowest terms, then a and b are relatively prime, so b won't divide a without remainder (unless b = 1 or b = -1). Exponentiation will not create any new prime factors! -- hz > Unfortunately mathematical reasoning isn't religion where "beliefs" > would be much relevant. Beliefs are not all religious. MoeBlee Nam D. Nguyen says... >Unfortunately mathematical reasoning isn't religion where "beliefs" >would be much relevant. Well, that's what *you* believe, but I don't. -- Daryl McCullough Ithaca, NY > Nam D. Nguyen says... > >> Unfortunately mathematical reasoning isn't religion where "beliefs" >> would be much relevant. > > Well, that's what *you* believe, but I don't. Well then, Jesus of Nazareth is the only Son of God is mathematically true? Or is that *false*, logically specking? > -- > Daryl McCullough > Ithaca, NY >> Nam D. Nguyen says... >> [quoted text clipped - 5 lines] >Well then, Jesus of Nazareth is the only Son of God is mathematically >true? Or is that *false*, logically specking? What in the world are you talking about? Oh, never mind. -- Daryl McCullough Ithaca, NY >>> Nam D. Nguyen says... >>> [quoted text clipped - 5 lines] > > What in the world are you talking about? I'm talking about religion kind of beliefs that you seem to believe relevant in mathematical reasoning. You seemed to understand that: "Well, that's what *you* believe, but I don't"! > Oh, never mind. So, you shouldn't pretend. > -- > Daryl McCullough > Ithaca, NY On Dec 17, 6:37 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 11 lines] > reason to believe that the negation of Goldbach's conjecture > is true. Why is a consistency proof of a theory in a stronger theory interesting? >On Dec 17, 6:37 am, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: [quoted text clipped - 16 lines] >Why is a consistency proof of a theory in a stronger theory >interesting? Didn't I just answer that? >> The *strength* of the system is not relevant so much as whether the >> axioms are themselves intuitively true. A proof in a theory whose >> axioms are intuitively true is more useful and interesting than >> a proof in a theory whose axioms are not intuitively true. -- Daryl McCullough Ithaca, NY Newberry says... >On Dec 12, 7:57 am, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> >> >b) No formal proof of PA has any cogency (TF explicitly admitted this) >> [quoted text clipped - 7 lines] > >You admitted this a few times on this thread. No, I did not. I never said that the proofs lack "cogency". >Torkel Franzen says "the soundness proof of PA is not intended >to allay any doubts at all." p. 110 That is not the same thing as saying that the proof "lacks cogency". >How do you know that ZFC + an axiom of infinity are consistent? Whether I "know" that or not depends on what you mean by "know". For practical purposes, you know something if you believe it and there are goods reasons for believing it and believing otherwise seems difficult. That's the case with ZFC. I don't have any absolute argument for why it is consistent, it's just that it seems very unlikely to me that it could be inconsistent and to have that inconsistency undiscovered before now. >> Whether a proof is "cogent" is a *psychological* fact. Are >> we convinced by it? There are convincing arguments for the [quoted text clipped - 3 lines] >consistency but you would be convinced if were proven by principles >that go beyond it? It depends on which principles you are talking about. If I accept the principles, then I accept the conclusions from them. >> What reason is there to say that? That's ridiculous. That's >> completely false. Almost *no* computer programs work by >> making formal proofs. Yet there are computer programs that >> arrive at mathematical conclusions. > >Examples? You're kidding, right? I can write a program using Newton's method to compute square roots. There is no theorem proving involved. Look at any mathematical package: Macsyma, Mathematica, etc. *None* of them use theorem proving. NASA uses computers to compute trajectories for rockets. Those programs don't use theorem proving. Handheld calculators do arithmetic. They don't do it by theorem proving. As far as I know, there are *no* practical computer programs that work by proving theorems. >> What you are saying is nonsense. > >So let's try the proof this way. Let us assume we are programmed as >ZFC with a strong axiom of infinity. I certainly don't believe that, but let's continue for the sake of argument. >So we can prove the consistency >of ZFC to ourselves. It depends on what you mean by a "strong axiom of infinity". But let's say ZFC + some large cardinal axiom. That would allow you to prove that ZFC is consistent. >But we also know that we do not know if ZFC with >a strong axiom of infinity is consistent. So what? Do you know that your reasoning is consistent? I believe that it's *not*. People make mistakes all the time. Human reasoning is most likely *not* consistent. We don't worry about it because our reasoning is self-correcting. If we discover an inconsistency, we just modify our beliefs to try to make them consistent again. The original Frege set theory was inconsistent. So what? All the important proofs from Frege set theory could be transferred to ZFC. Inconsistency is not the end of the world, and there is no good reason for us to believe that human reasoning is always consistent. >If it is not than the proof of ZFC's consistency is invalid. >So we are convinced about ZFC's consistency because ZFC with >a strong axiom of infinity is our horizont. I don't know the word "horizont". The reason we are convinced that ZFC is consistent is because we have an intuitively appealing way to think about ZFC in terms of the cumulative hierarchy. It's very difficult to imagine how it could possibly be inconsistent. But that's the extent of it, as far as I can see. There is no deep, infallible sense in which it is impossible for us to be wrong about the consistency of ZFC. >We are fooling ourselves. That's a drastic way of putting it. Lacking certainty is not the end of the world. People can live with uncertainty. People can live with inconsistency. >If do not want to admit that we are fools we have to drop the >assumption that we are equivalent to ZFC with a strong axiom >of infinity. So you are saying that you would find it *embarassing* to be equivalent to ZFC plus some large cardinal axiom. Maybe so. But that's not a proof that we're not. >How else can we possibly prove a consistency of a given theory T? Basically, there are two ways of proving consistency of a theory. 1. Show that the theory has a model, or 2. Show, by transfinite induction on proofs that it is impossible to prove a contradiction. >A theory could prove its own consistency. Not in any interesting or useful way. >What else? A heuristic learning program cannot do it because >mathematics is not an empirical science. I disagree. Some aspects of mathematics are empirical. The question of what axioms are useful, consistent and interesting is an empirical question. A heuristic learning program can discover interesting axioms. The second part of mathematics is nonempirical, which is determining what rigorously follows from axioms. That certainly is not beyond computers. >So we have >A) an inreasing hierarchy of theories,and we have proven in the first >paragraph that either we are NOT equivalent to any of those or that we >do not know if PA is consistent. You haven't proven anything at all. That's why I really wish that you would try to put your arguments in the form of a syllogism. Make your assumptions *explicit*. Your kind of fuzzy reasoning is incapable of actually determining what follows from your assumptions. -- Daryl McCullough Ithaca, NY >As far as I know, there are *no* practical computer programs >that work by proving theorems. The point you are making is of course correct; however, it is not true that there are no practical computer programs that work by proving theorems. The key words are "automated theorem proving." See for example http://www.cs.miami.edu/~tptp/OverviewOfATP.html for more information. SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences tchow@lsa.umich.edu says... >>As far as I know, there are *no* practical computer programs >>that work by proving theorems. [quoted text clipped - 4 lines] >example http://www.cs.miami.edu/~tptp/OverviewOfATP.html for more >information. I know that there are automated theorem proving programs (I have actually worked to build one, once upon a time) but I was not aware that they were used for anything very practical. (The one I worked on certain was not very practical.) -- Daryl McCullough Ithaca, NY On Dec 15, 11:17 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 146 lines] > Daryl McCullough > Ithaca, NY Let me see if I can summarize your position. a) The human neural system does not surpass the Turing machine. b) We are NOT equivalent to any theory PA, ZFC, ZFC + an axiom of infity, T_3, T_4 for any n c) We are programmed as a heuristic learning algorithm d) Mathematics is at least partially an empirical science Is this a fair characterization of your position? If it is not could you succintly summarize so we would not look a like a program stuck in an infinite loop? Newberry says... >Let me see if I can summarize your position. >a) The human neural system does not surpass the Turing machine. Yes, I believe that. >b) We are NOT equivalent to any theory PA, ZFC, ZFC + an axiom of >infity, T_3, T_4 for any n It depends on what you mean by being "equivalent". Humans don't *have* a fixed theory. We can be talked into accepting statements as true, but there is no good reason to think that we are perfectly consistent about it. Perhaps a human could be talked into believing the Continuum Hypothesis, but with a different argument, could be talked into believing its negation. Humans are not very usefully characterized by a formal theory. >c) We are programmed as a heuristic learning algorithm I would say that we *have* heuristic learning algorithms. I wouldn't say that we *are* those algorithms, or that we were *programmed* (unless you want to call natural selection a form of programming). >d) Mathematics is at least partially an empirical science Yes. >Is this a fair characterization of your position? Pretty good. -- Daryl McCullough Ithaca, NY On Dec 17, 6:54 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > > >Let me see if I can summarize your position. > >a) The human neural system does not surpass the Turing machine. > > Yes, I believe that. The same way you could prove that there is no free will. Do you also believe that? > >b) We are NOT equivalent to any theory PA, ZFC, ZFC + an axiom of > >infity, T_3, T_4 for any n [quoted text clipped - 22 lines] > > Pretty good. Let me make sure that I understand the whole thing. By the heuristic learning algorithms we have arrived at the conclusion that the axioms of ZFC are true and also that ZFC is consistent. (What is the degree of certainty, 100%?) In additions there is a also a consistency proof of ZFC in a stronger system (ZFC + an axiom of infinity. (What is the cogency of this proof?) Newberry says... >On Dec 17, 6:54 am, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: [quoted text clipped - 6 lines] > >The same way you could prove that there is no free will. I never said I could do that, did I? I'm not sure what free will really means, so I won't venture to say that there is or is not such a thing. >Let me make sure that I understand the whole thing. By the heuristic >learning algorithms we have arrived at the conclusion that the axioms >of ZFC are true and also that ZFC is consistent. (What is the degree >of certainty, 100%?) In additions there is a also a consistency proof >of ZFC in a stronger system (ZFC + an axiom of infinity. (What is the >cogency of this proof?) I don't know what you mean by "cogency". As I have said, a proof is only interesting if you learn something from it. If the conclusion is a nonobvious consequence of the axioms. It's a little bit interesting that the existence of large cardinals implies the consistency of ZFC, but it's not difficult to see why. The reason I believe that ZFC is consistent is certainly not because of large cardinal axioms, it's because I find the cumulative hierarchy picture of the construction of the set-theoretic universe to be compelling. -- Daryl McCullough Ithaca, NY On Dec 17, 6:54 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 29 lines] > > Pretty good. I think that c) and d) are absurd. Newberry says... >> >c) We are programmed as a heuristic learning algorithm >> [quoted text clipped - 12 lines] > >I think that c) and d) are absurd. Oh, well. -- Daryl McCullough Ithaca, NY On Dec 15, 8:17 pm, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > > >A theory could prove its own consistency. > > Not in any interesting or useful way. First of all, I have to say how much I admire your patience. Aren't you ever going to get tired? Secondly, are any proofs of PA's consistency "interesting or useful"? Or is it only when a theory proves its own consistency that it is not interesting or useful? >Secondly, are any proofs of PA's consistency "interesting or useful"? Well, there's Gentzen's proof using primitive recursive arithmetic plus quantifier-free transfinite induction up to epsilon_0. This is interesting, because in PA one can carry out transfinite induction up to any ordinal strictly less than epsilon_0, so this shows that epsilon_0 captures exactly the proof-theoretic strength of PA. SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences On Dec 16, 4:49 am, tc...@lsa.umich.edu wrote: > In article <9438634d-bc9c-4575-80d6-4351b1d50...@18g2000hsf.googlegroups.com>, > [quoted text clipped - 5 lines] > up to any ordinal strictly less than epsilon_0, so this shows that > epsilon_0 captures exactly the proof-theoretic strength of PA. That isn't the point of what was a two-part question, whose second part you snipped. I take "interesting" to be psychological, so what may be interesting to you may not be interesting to me, so answering one part is of no value (IMHO). Daryl said that a theory cannot prove its own consistency in "any interesting or useful way." I'm curious then whether he holds there are proofs (such as Gentzen's) of the consistency of PA which he holds to be "interesting or useful", and why he excludes such interest or usefulness from proofs by theories of their own consistency. > On Dec 16, 4:49 am, tc...@lsa.umich.edu wrote: > [quoted text clipped - 18 lines] > to be "interesting or useful", and why he excludes such interest or > usefulness from proofs by theories of their own consistency. Yes, he holds that consistency proofs of PA in a stronger system are interesting and useful. > > On Dec 16, 4:49 am, tc...@lsa.umich.edu wrote: > [quoted text clipped - 21 lines] > Yes, he holds that consistency proofs of PA in a stronger system are > interesting and useful. Again,that's only half of it. I'm curious, for a constant meaning of "interesting and useful", whether he holds that consistency proofs of PA in a stronger system are interesting and useful, while a theory proving itself consistent cannot be "interesting and useful", and what would explain this difference. > Daryl said that a theory cannot prove its own consistency in "any > interesting or useful way." I'm curious then whether he holds there > are proofs (such as Gentzen's) of the consistency of PA which he holds > to be "interesting or useful", Of course he does. > and why he excludes such interest or usefulness he doesn't. > from proofs by theories of their own consistency. Because G2IT proves that all the theories (in this realm, satisfying the hypotheses of G2) that prove their own consistency ARE INCONSISTENT, THAT'S why. WE ALL exclude "interest or usefulness" from proofs in inconsistent theories. > > Daryl said that a theory cannot prove its own consistency in "any > > interesting or useful way." I'm curious then whether he holds there [quoted text clipped - 14 lines] > THAT'S why. WE ALL exclude "interest or usefulness" from proofs in > inconsistent theories. So there is an implicit qualification that I missed? Daryl is only talking about theories of a particular type, whereby proving their own consistency is equivalent to the theory being inconsistent? Well, then I understand why they are not interesting or useful, but I missed the qualification. abo says... >On Dec 15, 8:17 pm, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: [quoted text clipped - 10 lines] >Or is it only when a theory proves its own consistency that it is not >interesting or useful? Any proof is only interesting (in my opinion) if the conclusion is not obviously a consequence. So a proof of Phi from a theory that has Phi as an axiom is pretty uninteresting. The fact that ZFC is provably consistent in the theory ZFC + Con(ZFC) is pretty uninteresting. But the proof that ZF + the negation of the axiom of choice is provably consistent in the theory ZFC + Con(ZFC) is interesting. -- Daryl McCullough Ithaca, NY On Dec 17, 3:45 pm, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > abo says... > [quoted text clipped - 27 lines] > > - Show quoted text - So what is the reason why you exclude a proof (in T) that T itself is consistent from being interesting and useful? Is it because you are talking implicitly of a theory of a certain type, where Godel's 2nd Theory applies? (And so T would be inconsistent?) > Secondly, are any proofs of PA's consistency "interesting or useful"? Sure. From Gentzen's consistency proof we learn, for example, that if PA proves a statement of the form "AxEyP(x,y)" with P decidable, there is a function F, below the epsilon-0'th level of a hierarchy of fast growing recursive functions, such that AxP(x,F(x)). > Or is it only when a theory proves its own consistency that it is not > interesting or useful? Given that inconsistent theories prove their own consistency, if we merely know of some theory T that it proves its own consistency nothing interesting at all can be concluded. If, inspecting the proof, we find that only principles we accept as correct are in fact used in the proof, its then of course useful in convincing us of the consistency of T. For most theories we're interested we know this can't happen, since the conditions of the second incompleteness theorems apply. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus On Dec 19, 2:40 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi> wrote: > > Secondly, are any proofs of PA's consistency "interesting or useful"? > [quoted text clipped - 9 lines] > know of some theory T that it proves its own consistency nothing interesting > at all can be concluded. That's pretty vacuous. Given that inconsistent theories prove "PA is consistent", if we merely know of some theory T that it proves "PA is consistent" nothing interesting at all can be concluded. > If, inspecting the proof, we find that only > principles we accept as correct are in fact used in the proof, its then of > course useful in convincing us of the consistency of T. For most theories > we're interested we know this can't happen, since the conditions of the > second incompleteness theorems apply. Yes, well I was interested in Daryl's definitive statement that no such proof could be interesting or useful. Apparently he meant it only to apply to theories where the conditions of the incompleteness theorem 2 apply, in which case I can understand why he thinks that there are proofs of "PA is consistent" which are interesting and useful, but not proofs (in T) of "T is consistent." > Given that inconsistent theories prove "PA is consistent", if we > merely know of some theory T that it proves "PA is consistent" nothing > interesting at all can be concluded. Right. In general, merely from the fact that this-or-that theory proves something, nothing interesting at all can be concluded. In case of particular theories immensely interesting mathematical knowledge can sometimes be obtained on knowing that P is provable in the theory, and even more information extracted from a particular proof, e.g. upper or lower numeric bounds on this-or-that. (That's one of the concerns of modern proof theory, after all.) > Yes, well I was interested in Daryl's definitive statement that no > such proof could be interesting or useful. Apparently he meant it > only to apply to theories where the conditions of the incompleteness > theorem 2 apply, in which case I can understand why he thinks that > there are proofs of "PA is consistent" which are interesting and > useful, but not proofs (in T) of "T is consistent." That would be a reasonable reading of his comments. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus > Whether I "know" that or not depends on what you mean by "know". > For practical purposes, you know something if you believe it and [quoted text clipped - 3 lines] > to me that it could be inconsistent and to have that inconsistency > undiscovered before now. Why? We don't have any good data on which to judge the likelihood of that. Happily, as you later note, we have much better grounds to think ZFC is consistent, sound for arithmetical statements, and so on. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus Aatu Koskensilta says... >> Whether I "know" that or not depends on what you mean by "know". >> For practical purposes, you know something if you believe it and [quoted text clipped - 7 lines] >Happily, as you later note, we have much better grounds to think ZFC is >consistent, sound for arithmetical statements, and so on. The reasons I personally believe that ZFC is consistent is because of the intuitively clear picture of the set-theoretic universe given by the cumulative hierarchy. It's very hard to imagine what could go *wrong*. -- Daryl McCullough Ithaca, NY > The reasons I personally believe that ZFC is consistent is because > of the intuitively clear picture of the set-theoretic universe given > by the cumulative hierarchy. It's very hard to imagine what could > go *wrong*. Right. I was just noting that the "we haven't yet run into an inconsistency" argument is a very weak one, and that, in fact, as you did note, we do have a much better reason for believing ZFC consistent, sound for arithmetical statements, and so on. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus "Newberry" <newberryxy@gmail.com> ha scritto >> That's a ridiculous thing to say. That's completely >> false. I can program a machine to play games, process [quoted text clipped - 3 lines] > Come on! Obviously I meant that machines cannot arrive at arithmetic > result by any other means than formal proofs. What about neural networks? > "Newberry" <newberr...@gmail.com> ha scritto > [quoted text clipped - 7 lines] > > What about neural networks? I believe they are equivalent to Turing computability. "Newberry" <newberryxy@gmail.com> ha scritto >> >> That's a ridiculous thing to say. That's completely >> >> false. I can program a machine to play games, process [quoted text clipped - 7 lines] > > I believe they are equivalent to Turing computability. Yes but they don't follow axioms and inference rules, so how could you consider the process of a neural network to be a "formal proof"? Newberry says... >> "Newberry" <newberr...@gmail.com> ha scritto >> [quoted text clipped - 9 lines] > >I believe they are equivalent to Turing computability. That's not the question. The question was whether they work via formal proofs. They clearly do not. So it is false to say that "machines arrive at arithmetic results by formal proofs". Now, what is the case is this: For any computable way to arrive at true arithmetical results, there is a formal theory such that those results are formally derivable from the axioms of the theory. In that sense, anything a computer can do in arithmetic is no more powerful than theorem proving. But by that definition, nothing a *human* can do is more powerful than theorem proving. That's what I meant in accusing you of using a "double standard". You take a single concept, such as "being equivalent to a formal theory", and you use two *different* definitions of the phrase. You apply one definition to computers, and a *different* definition to humans, and you come up with the conclusion that the phrase applies to computers but not to humans. That's blatantly *invalid* reasoning. If you want to prove that humans can do something that no computer can, you must use a *consistent* definition of your terms. The sense in which all computer computations are essentially the same as theorem proving is the same sense in which all *human* mathematical reasoning is essentially the same as theorem proving. -- Daryl McCullough Ithaca, NY On Dec 13, 6:42 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 22 lines] > powerful than theorem proving. But by that definition, nothing > a *human* can do is more powerful than theorem proving. You seem to believe that humans are basically neural networks. If that is your position you should have stated it explicitly. > That's what I meant in accusing you of using a "double standard". > You take a single concept, such as "being equivalent to a formal [quoted text clipped - 5 lines] > that no computer can, you must use a *consistent* definition of > your terms. There is plenty of wrong with what you are saying but the main one is this: if we first prove P from certain assumptions and then we prove ~P does it mean that we have refuted P? You seem to believe that humans are neural networks and therefore equivalent to machines/ computers. Hence you think it must be wrong to claim that humans surpass any machine. But I did not say that. I said there were possibilities 1), 2), 3). If you refute 2) that leaves 1) and 3). I told you this about three months ago. It is not certain that human brains are equivalent to our model of neural networks that are Turing compatible. You do not know what is going on inside the neurons. For example Penrose speculated something about the qunatum wave collapsing and entering the consciousness. Before somebody starts arguing about Penrose - the point is that we do not have any idea how the biological neural networks function. So I do not know if this the way to refute 2). > The sense in which all computer computations are essentially the > same as theorem proving is the same sense in which all *human* [quoted text clipped - 3 lines] > Daryl McCullough > Ithaca, NY Newberry says... >You seem to believe that humans are basically neural networks. I didn't say that. Someone else asked about neural networks. My point is just that the argument from Godel's theorem to the conclusion that humans surpass computer programs is simply not a valid argument. >> That's what I meant in accusing you of using a "double standard". >> You take a single concept, such as "being equivalent to a formal [quoted text clipped - 9 lines] >this: if we first prove P from certain assumptions and then we prove >~P does it mean that we have refuted P? It depends on what you mean by "refute". You have to establish clearly what you mean by words before you can hope to prove things about them. That's the problem with your argument is that you don't clearly explain what your terms mean, and the meaning shifts in the middle of your argument. That's a logical fallacy. >You seem to believe that >humans are neural networks and therefore equivalent to machines/ >computers. I didn't say anything about neural networks. My argument that human mathematical reasoning is no more powerful than a computer is simple: Humans have finite memories, finite lifetimes, finite processing speed. Therefore, the number of statements that we could ever come to comprehend, let alone prove, is finite. Any finite set is computable. Therefore, human mathematical abilities do not extend beyond what is computable. QED. >Hence you think it must be wrong to claim that humans >surpass any machine. No, what I think is that your argument in favor of humans surpassing machines is bogus. It's sloppy thinking. If you actually tried to put it in the form of a syllogism in which the conclusions *logically* follow from the premises, you would see how silly your argument is. But by being sufficiently sloppy, you hide your arguments' flaws (at least from yourself). >But I did not say that. I said there were >possibilities 1), 2), 3). But that's false. Those aren't the only three possibilities. The claim that those are the only three possibilities is itself the result of sloppy thinking on your part. I reject all three possibilities 1. Humans have no idea whether PA is consistent. That's false. We have good empirical and analytic evidence that it is consistent, and all that evidence is equally accessible to a computer program. 2. There is a formal theory that can prove its own consistent. Godel's theorem suggests that that is false. It's possible that there is some odd way to interpret the claim so that it can be made true. 3. Humans surpass machines. There is a good argument that that is false. So I reject all three of your options. -- Daryl McCullough Ithaca, NY -- Daryl McCullough Ithaca, NY >No, it's not an accident. PA is a generalization and >systematization of human experience with natural numbers >over thousands of years. It's a very natural thing to >come up with if one wants to formalize mathematics. >A machine with heuristics for coming up with axiomatizations >of mathematics would come up with the same axioms. Again this may be a quibble, but I don't think I believe this. Peano, for example, did not come up what is now called PA. Restricting the induction axiom to all *first-order formulas* is by no means an obvious thing to do. SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences tchow@lsa.umich.edu says... >>No, it's not an accident. PA is a generalization and >>systematization of human experience with natural numbers [quoted text clipped - 6 lines] >example, did not come up what is now called PA. Restricting the induction >axiom to all *first-order formulas* is by no means an obvious thing to do. Sorry for being a little loose here. The argument here is really about some simple theory of mathematics that is sufficient to prove lots of stuff. "Peano Arithmetic" is a stand-in for such a theory. If you think that there are nonobvious parts to PA, then we can switch to something even more basic, such as Robinson arithmetic. Basically something like 1. x+0 = x 2. x+(y+1) = (x+y)+1 3. x*0 = 0 4. x*(y+1) = (x*y) + x 5. x+1 is unequal to 0 6. x+1 = y+1 -> x=y I guess I would agree that the principle of induction is nonobvious. But Robinson arithmetic is already strong enough that you can't prove its consistency without going to a more powerful theory (whose consistency requires an even more powerful theory, etc.) But as for the Peano axioms, why *is* it restricted to first order formulas? As opposed to what? Second-order formulas? Does some kind of inconsistency result from applying induction to higher-order formulas? -- Daryl McCullough Ithaca, NY On Dec 11, 5:22 pm, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > tc...@lsa.umich.edu says... > [quoted text clipped - 34 lines] > formulas? Does some kind of inconsistency result from > applying induction to higher-order formulas? You can have second-order PA just as well as first-order PA. The mathematical differences: induction in second-order PA can be an axiom rather than a schema, or if it is a schema it has more instances (because second-order PA has more formulas); and usually the addition and multiplication axioms are supressed, since addition and multiplication can be defined in terms of the successor function in second-order PA. > ... > But as for the Peano axioms, why *is* it restricted to > first order formulas? As opposed to what? Second-order > formulas? And higher. Really interesting history here, though I don't know it as well for PA as for ZF, though I suspect the reasons are largely the same in both cases. The original dispute seems to have been between Skolem and Zermelo over how to formally express the meaning of "definite property" in the formulation of the axiom schema of Separation -- Skolem arguing for FOL and Zermelo pushing a gonzo higher-order infinitary logic over worries about the essential non-categoricity and incompleteness of first-order theories. Greg Moore's work covers the history very well here, notably: Moore, G. H. (1980). Beyond first-order logic: The historical interplay between mathematical logic and axiomatic set theory. History and Philosophy of Logic, I, 95-137. Moore, G. H. (1988). The emergence of first-order logic. In Aspray and Kitcher (eds), History and Philosophy of Modern Mathematics, Univ of Minnesota Press, 95-135. See also Eklund, M. (1996). On how logic became first-order. Nordic Journal of Philosophical Logic, I(2), 147-167. Van Dalen, D., and Ebbinghaus, H. (2000). Zermelo and the Skolem paradox. The Bulletin of Symbolic Logic, 6(2), 145-161. Eklund's piece is available on the web. > > ... > > But as for the Peano axioms, why *is* it restricted to [quoted text clipped - 28 lines] > > Eklund's piece is available on the web. at http://www.hf.uio.no/ifikk/filosofi/njpl/vol1no2/howlogic/howlogic.pdf Haven't read the above yet, but see also: Jose Ferreiros (2001). The Road to Modern Logic -- An Interpretation. The Bulletin of Symbolic Logic, 7(4), 441-484. at http://www.math.ucla.edu/~asl/bsl/0704/0704-001.ps -- hz > See also > > Eklund, M. (1996). On how logic became first-order. Nordic Journal of > Philosophical Logic, I(2), 147-167. > Eklund's piece is available on the web. It concludes with the following: > In his book The Concept of Logical Consequence (1990), > John Etchemendy discusses Tarski's analyses of the concepts > of logical truth and logical consequence very critically. > Etchemendy's contention is that the analyses are faulty: > they are not even extensionally correct. Has that contention met with any general respect? It would seem to be far more iconoclastic than what AK calls my "idiosyncratic" doubts about the "truths" of first- order arithmetic. > Today, an influential criticism of second-order logic is that > within second-order logic, sentences which should not > come out logically true or false come out that way. I'm sure I'm not going to respect that. > For example, the sentence of pure second-order logic > which expresses the continuum hypothesis itself comes > out logically true or false, respectively, depending on its > truth or falsity in the underlying set theory. The underlying set theory CANNOT PROVIDE any truth or falsity. MODELS provide truth or falsity. The underlying set theory would have to THEORETICALLY PROVE /DISprove/DECIDE CH in ORDER to be able to be supplying a truth-value for it. > So if we hold that the continuum hypothesis is not > logically true or false, second-order logic is, > qua logic, seriously deficient. That has all kinds of carts before all kinds of horses. >> In his book The Concept of Logical Consequence (1990), >> John Etchemendy discusses Tarski's analyses of the concepts [quoted text clipped - 6 lines] >calls my "idiosyncratic" doubts about the "truths" of first- >order arithmetic. See http://alum.mit.edu/www/tchow/etchemendy_review.pdf for a review of Etchemendy's book. Etchemendy is concerned with the right way to analyze the concept of logical consequence, and is not claiming that what mathematicians regard as "truths" are not really true. SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences On Dec 14, 11:02 am, tc...@lsa.umich.edu wrote: > Etchemendy is concerned with the right way to analyze the concept of logical > consequence, and is not claiming that what mathematicians regard as "truths" > are not really true. You have just fallen headlong into the yawning abyss of false distinctions. The point is that these issues are not separable. The Tarskian outlook makes truth model-dependent. It also defines logical consequence model- theoretically. > The Tarskian outlook makes truth model-dependent. Really? That sounds like a confusion of Tarski's 1936 definition of truth with his 1950s definition of the different notion of truth-in-a- model. > Really? That sounds like a confusion of Tarski's 1936 definition of > truth with his 1950s definition of the different notion of truth-in-a- > model. No, really, it doesn't. It is almost impossible to confuse the two since they are occurring in two different contexts. If you start with a signature and a set of axioms phrased in the (it will be unique) first-order language over that signature, then the question of which sentences in that language "are true" IS WHOLLY model-dependent. Any non-model dependent definition is simply not going to apply, or if it does, it will (if it is coming from the same author and the same perspective) apply in a way that is entirely harmonious with the model-theoretic perspective. > If you start with a signature and a set of axioms phrased > in the (it will be unique) first-order language over that > signature, then the question of which sentences in that language > "are true" IS WHOLLY model-dependent. In saying that the FOL with a given signature is unique (I take it you mean up to boring relabelling of predicates etc) suggests that you are thinking of a formalized language as a purely syntactic system. Why? I know that logic books occasionally talk like this, but this has always seemed to me simply to be a misuse of the word "language". A formalized language in which we do some mathematics is supposed to be a *language* in which we can communicate some mathematical truths (an uninterpreted syntax communicates nothing): so such a formalized language -- as much a contentful language as English or Polish but better behaved -- is much better thought of as an ordered pair of syntactic system and an interpretation. And on that understanding, it makes perfect sense of course to talk of a sentence of a formalized language as true, tout court. >> If you start with a signature and a set of axioms phrased >> in the (it will be unique) first-order language over that [quoted text clipped - 4 lines] > mean up to boring relabelling of predicates etc) suggests that you are > thinking of a formalized language as a purely syntactic system. Every language (e.g. ordinary language, "street" language, undergraduate-student language, formalized language, etc...) is syntactical of course. The difference is formalized language would follow 100% rigidity of grammar and otherwise *rules* of formula transformation or mapping, in a "conversation" (a.k.a. reasoning). > Why? Because, *up to a point*, we couldn't tell what the ordinary language might have meant. "Ordinary" language is way *too intuitive* to be trusted in complex reasoning. > I know that logic books occasionally talk like this, but this has always > seemed to me simply to be a misuse of the word "language". I'd think it's the other way around: without the strict no-emotion-no-interpretation rigidity of the syntactical rule of formalized language, it'd be only a matter of time before "reasoning" would be misused. > A formalized language in which we do some mathematics is supposed to be > a *language* in which we can communicate some mathematical truths (an > uninterpreted syntax communicates nothing): That really depends on what we mean by "*some* mathematics" or "communicate [...] mathematical truths", or "mathematical truths". To most of us, "house" means "home"; but to a Zen Master, "home" doesn't necessarily mean "house"! (Seriously, from the ordinary arithmetic perspective ~(1+1=0) is true, but from a modulo arithmetic perspective it could be the other way around! So what would "add" really mean, in a language where there can be no misuse?) > so such a formalized language -- as much a contentful language > as English or Polish but better behaved -- is much better thought > of as an ordered pair of syntactic system and an interpretation. The problem here is there is always *a pair* of interpretations: one would negate the other! So which one should we love enough to crown to the "syntactical-system chair", or hate enough to dethrone from it? > And on that understanding, it makes perfect sense of course to talk > of a sentence of a formalized language as true, tout court. On this understanding then it'd not seem to make perfect sense at all. > In saying that the FOL with a given signature is unique (I take it you > mean up to boring relabelling of predicates etc) suggests that you are > thinking of a formalized language as a purely syntactic system. Why? Probably because it says "cs" (before ".unc.edu) in my eddress. I > know that logic books occasionally talk like this, but this has always > seemed to me simply to be a misuse of the word "language". We truly couldn't care less how this seems to you. Over here it is entirely normal to talk about a hierarchy of formal languages (regular, context-free, etc.). ALL these things are NOTHING BUT sets of strings. Over here, "a language is a set of strings" IS ANALYTIC. You *can*Not argue with A DEFINITION. > A formalized language in which we do some mathematics is supposed to be SHUT *UP* !! JEEzus! For Whom (capital w very much necessary in your usurped role) have you MISTAKEN yourself that you think you GET to say what MATH is supposed to be?? A formal language is a set of strings. NO, *you*, personally, DO NOT *get* to affreight it with more of your favorite baggage. > > know that logic books occasionally talk like this, but this has always > > seemed to me simply to be a misuse of the word "language". [quoted text clipped - 4 lines] > are NOTHING BUT sets of strings. Over here, "a language is a set > of strings" IS ANALYTIC. You *can*Not argue with A DEFINITION. I wasn't "arguing with a definition"; but it is a somewhat misleading definition, since uninterpreted syntactic systems in themselves convey no messages, have no semantic content, can't be used to communicate anything. And yet you might well suppose that it is ANALYTIC of the ordinary notion of a language that it is a means for communication. (If you do insist on using "formal language" for uninterpreted strings, then we need to keep reminding ourselves that a formal language in that sense is no more a language than a toy gun is a gun.) > > A formalized language in which we do some mathematics is supposed to be > > SHUT *UP* !! > JEEzus! For Whom (capital w very much necessary in your usurped role) > have you MISTAKEN yourself that you think you GET to say what MATH > is supposed to be?? Oh do calm down. I didn't saying what maths is supposed to be. What I wrote was "A formalized language in which we do some mathematics is supposed to be a *language* in which we can communicate some mathematical truths". Obviously, in your sense of formal language, where formal languages have no content, a formal language cannot be used as a vehicle to do mathematics. Fair enough. Such objects may be interesting but not what I was talking about. I was talking about contentful formalized languages, such as the languages of Principia, or of first-order PA, or ZFC, meaningful languages in which we *can* do more or less mathematics. > I wasn't "arguing with a definition"; but it is a somewhat misleading > definition, since uninterpreted syntactic systems in themselves convey > no messages, have no semantic content, can't be used to communicate > anything. What UTTER bullshit. You do NOT have to affix a whole theory of semantics, OR interpret anything, IN ORDER to allege that SOME of the strings in the language ARE AXIOMS! THAT ALLEGATION (and if you are going to finish alleging it in finite time, then the subset-of-the-strings-so-blessed had DANGwell better be recursive) is how you cause the communication. THE AXIOMS are what you communicate with. The axioms are characterizable AS such SYNTACTICALLY, after the signature has been established. And they communicate just fine, thank you. > What UTTER bullshit. You do NOT have to affix a whole theory > of semantics, OR interpret anything, IN ORDER to allege that > SOME of the strings in the language ARE AXIOMS! That's trivially true of course. > THE AXIOMS are what you communicate > with. The axioms are characterizable AS such SYNTACTICALLY, > after the signature has been established. And they communicate > just fine, thank you. No they don't, if all I give you is the signature. I give you a formal syntax with the signature of the language of PA. I set down an axiom x + 0 = 0 That communicates *nothing* unless I also tell you whether '+' expresses e.g. addition or multiplication, and '0' denotes e.g. zero or one (or maybe something else altogether). > > The axioms are characterizable AS such SYNTACTICALLY, > > after the signature has been established. And they communicate > > just fine, thank you. > > No they don't, if all I give you is the signature. Well, by definition, the signature WAS NOT all; we also gave THE AXIOMS. > I give you a formal syntax with the signature of the language of PA. Not necessarily. You had said something earlier about "up to relabeling of predicates". The truth is that it does not even matter what the signature is. The axioms are what matter, and, in fact, even THEY could be phrased IN ANY signature, as long as it had the right numbers of functions of the right arities. There really is a sense in which the (first-order) theory of equivalence and the (first-order) theory of membership have THE SAME signature (one binary predicate) -- the fact that it is spelled = in one case and e in the other is not even important except insofar as we often have to use both of them in the same paper, and therefore need a way to tell them apart. It also helps to have a standard name simply because that spares you the burden of actually repeating/reproducing all the axioms, which are the only OTHER way to clarify what the predicate MEANS. > I set down an axiom > [quoted text clipped - 3 lines] > expresses e.g. addition or multiplication, and '0' denotes e.g. zero > or one (or maybe something else altogether). Dear educated FOOL: THE ONLY way you COULD POSSIBLY tell me that is with *MORE* A X I O M S . *sh.t*. How in the HELL are you going to condemn ONE SINGLE axiom for not "meaning" anything when it usually occurs in a context of INFINITELY many axioms??? It is the infinity TOGETHER that means something! The meaning of ANY individual axiom is AFFECTED by whether OTHER things that also use the same-symbols-occurring-in-the-axiom are or are not ALSO asserted CONJUNCTIVELY to be axioms ALONG WITH it, in the SAME axiom-set! > I give you a formal syntax with the signature of the language of PA. I > set down an axiom [quoted text clipped - 4 lines] > expresses e.g. addition or multiplication, and '0' denotes e.g. zero > or one (or maybe something else altogether). The something-else-altogether part is ENTIRELY important. I am NOT disagreeing with anything you say in this paragraph. We do not have to get argumentative or tendentious about THIS point. The argument is over what you think this point MEANS. The point is, even after you add THE ENTIRE INFINITY OF ALL THE REST of the Peano axioms TO this one, it REMAINS the case that you have no idea what 0 denotes or what + denotes. It REMAINS the case that these could be ANYthing. The point is that IT DOES NOT *MATTER* what they denote! NOBODY GIVES A *f.ck* what they "denote"! They don't NEED to denote ANYthing! The POINT is, whatEVER they denote, it will be A MODEL of these axioms! and the THEOREMS that follow from these axioms will be true ABOUT ALL the things that this ENTIRELY RADICALLY "ambiguous" or multi- denotative language might be about, SIMULTANEOUSLY. This is sort of a difference between don't-know and don't-care non- determinism. Your problem is that you are treating it as some kind of criticism or flaw that there is don't-know non-determinism about what the axioms are referring to. That is unpardonable. The actual fact of the case is that there is don't-care non- determinism and that this is AN ASSET, not a liability. > > I give you a formal syntax with the signature of the language of PA. I > > set down an axiom [quoted text clipped - 16 lines] > denote! > NOBODY GIVES A *f.ck* what they "denote"! Really? This person does care :-) Sure, it doesn't matter what e.g. the denoting expressions of the language of Peano Arithmetic mean when we are interested in certain questions -- e.g. which wffs follow from the axioms. It does matter if we are interested in certain other questions -- like what the theorems say about the natural numbers (which is, after all, why we are interested in the formalized version of everyday arithmetic -- we want to know more about the same subject matter that everyday arithmetic is about).. > Sure, it doesn't matter what e.g. the denoting expressions of the > language of Peano Arithmetic mean when we are interested in certain > questions -- e.g. which wffs follow from the axioms. The burden of proof is on you to explain why anybody should ever legitimately be interested in anything else, at least in THIS room. It does SAY "sci.logic" on the door. EVERYthing that is interesting in HERE is interesting in virtue of logic having made it so. > It does matter if > we are interested in certain other questions -- Except we're not. The question you are about to cite below IS NOT "other". > like what the theorems say about the natural numbers The theorems BY DEFINITION say THE SAME thing ABOUT ALL the models, since ALL the models of the axioms AGREE on the truth-values of EVERY sentence that is a theorem or the denial of one. EVERY theorem says something COMPLETELY CLEAR about the naturals, since the naturals ARE a model of the theory (the theory here is first-order PA). > (which is, after all, why we are interested in the formalized version > of everyday arithmetic -- No, No -- why we WERE interested. Why HILBERT was interested. GODEL shows that this is just a hopeless conceit. Not only is first-order PA inadequate, first-order ANYthing MUST fail to get SOME of this "right". The point is simply that naturals ARE INHERENTLY NOT characterizABLE by any finitary first- order framework. But that does not imply that the things that ARE first-order theorems (in ANY first-order theory of arithmetic) are any less "about" the naturals. Everything that is decided by the r.e. theory is decided BOTH about the naturals AND about every other possible model as well. It is the sentences that are NOT decided by the theory, that are NOT part of the theory, that may be ambiguous, in reference, in a BAD way. > we want to know more about the same subject matter > that everyday arithmetic is > about. The whole point about first-order PA is that you do not NEED to PRE-require that the theory be ONLY about that. Once you have the theory, ALL its theorems are AUTOMATICALLY about BOTH that AND all the other previously-undiscovered models. The NON-theorems (and non-denials of theorems, the undecided statements), however, are problematic. The problem is that we think we have this known denumerable set and that we can write a first-order generalization over it. That we factually can't is a surprising disappointment. It's extremely counter-intuitive. I was thinking that Etchemendy's alternative to Tarskian "interpretational" semantics might cure this. But my POINT was, as LONG as Tarskian-interpretational REMAINS the default semantics, blessing 1 model as standard and deprecating all the others simply is not defensible, unless you can generalize the criteria. > > It does matter if > > we are interested in certain other questions -- [quoted text clipped - 4 lines] > No, No -- why we WERE interested. Why HILBERT was interested. > GODEL shows that this is just a hopeless conceit. Does he? I think he'd be very surprised to hear that! We start off working in everyday mathematics. Wanting maximal rigour and absolute clarity, we reflect on our practices, and regiment our informal mathematical language, and regiment chunks of our everyday mathematics into nicely disciplined axiomatic systems. We construct, for example, the regimented theory most of us call first-order PA. This theory is as semantically contentful as the informal inchoate theory we started off with, just better disciplined. Of course, once that regimented theory is on the table, we can establish various things in it, and various things about it. Among the latter, we can show e.g. that it is not categorical, i.e. we can reinterpret the theory as having a quite different content which still makes true all the theorems. And one way of showing that is via Godel's incompleteness theorem plus Godel's completeness theorem. But none of that, as it were, saws off the branch we were sitting on and the possibility of such reinterpretation doesn't show that the regimented theory isn't just that, a genuine *theory* with the same content as (part of) our informal mathematics. > We start off working in everyday mathematics. You don't even know what that means. You are beginning by presuming the existence of an intended model. That is just not the way it's generally done any more. > Wanting maximal rigour > and absolute clarity, we reflect on our practices, and regiment our > informal mathematical language, and regiment chunks of our everyday > mathematics into nicely disciplined axiomatic systems. That is the BEGINNING, NOT THE END, of the process. > We construct, > for example, the regimented theory most of us call first-order PA. Almost nobody more than 10 years younger than you ever did any such thing. The rest of us STARTED with PA. It embodies the basic things we know, at first-order, about this realm. > This theory is as semantically contentful as the informal inchoate > theory we started off with, No, it isn't, and more to the point, PURELY BY VIRTUE OF BEING a first-order syntactic theory, IT IS NOT semantically contentful AT ALL. The completeness theorem is fundamentally a proof that first-order semantics SIMPLY DOESN'T EXIST. >Almost nobody more than 10 years younger than you ever did any >such thing. The rest of us STARTED with PA. It embodies the basic >things we know, at first-order, about this realm. George, you've *got* to be kidding here. Do you not have any clue how idiosyncratic your viewpoint is? Name one person other than yourself who "STARTED with PA." I've seen lots of "new math" movements, but never have I heard of anyone teaching arithmetic to five-year-olds by starting with PA. SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences > >Almost nobody more than 10 years younger than you ever did any > >such thing. The rest of us STARTED with PA. It embodies the basic > >things we know, at first-order, about this realm. On Dec 18, 2:38 pm, tc...@lsa.umich.edu wrote: > George, you've *got* to be kidding here. Do you not have any clue how > idiosyncratic your viewpoint is? Name one person other than yourself > who "STARTED with PA." > > I've seen lots of "new math" movements, but never have I heard of anyone > teaching arithmetic to five-year-olds by starting with PA. At some point at an age not MUCH later than that one, YOU (yes, YOU, Tim Chow) were taught that adding 0 to something does not change it. You were taught that 5+0=5. You were also taught that 5 was not special in this regard, and that EVERY natural number was like this. The fact that you were not told that "This is an axiom of PA" did NOT STOP you from HAVING, FACTUALLY, ACTUALLY been taught ONE axiom of PA, namely Ax[x+0=x]. The fact that it wasn't phrased specifically that way and that PA as an entity was not mentioned IS NOT relevant. You were also taught that Ax[x*1=1] from the FIRST day you were taught any multiplication tables. I repeat, style of spelling this stuff IS NOT relevant. The content of the axioms is what is relevant. > > >Almost nobody more than 10 years younger than you ever did any > > >such thing. The rest of us STARTED with PA. It embodies the basic [quoted text clipped - 25 lines] > NOT relevant. > The content of the axioms is what is relevant. They teach the induction schema to little kids? MoeBlee > They teach the induction schema to little kids? Nevermind. I see the point was ceded in another post. MoeBlee > The content of the axioms is what is relevant. Indeed. I thought that was my Fregean point, against your Hilbertian formalism, namely that the axioms do come with semantic content :-) > > The content of the axioms is what is relevant. > > Indeed. I thought that was my Fregean point, against your Hilbertian > formalism, namely that the axioms do come with semantic content :-) No, sorry. If the axioms are first-order then semantics is simply irrelevant. That is the GODELIAN point. That is the point that follows from the completeness theorem. More to the point, formalism is not opposed to content. Content by definition IS MERELY fancy form. On Dec 18, 2:38 pm, tc...@lsa.umich.edu wrote: > George, you've *got* to be kidding here. Do you not have any clue how > idiosyncratic your viewpoint is? Name one person other than yourself > who "STARTED with PA." Well, people in general, regardless of age, certainly don't start thinking about the totality of models of some axioms. The intended model that most of us started with thought that "the numbers" were finite digit-strings, with digit=={0..9}. Well, actually, we were maybe taught Roman numerals so that we would know that digit-strings were numerals rather than numbers, but that is actually a distinction that does NOT even matter; numerals are enough LIKE numbers to serve all the same purposes; the important point was just to remind people that numeration systems were possible as OPPOSED to necessary. Axiomatizations too, for that matter. On Dec 18, 2:38 pm, tc...@lsa.umich.edu wrote: > George, you've *got* to be kidding here. Do you not have any clue how > idiosyncratic your viewpoint is? Name one person other than yourself > who "STARTED with PA." Started DOING WHAT? We *start* by CALCULATING. That is *0th*-order. I am trying to remember what the FIRST first-order results I ever learned or PROVED were. I would say that the ones I learned first were identities. Those ARE axioms of PA *even* when they are not presented as such. In defense of your point I would say that I was next taught commutativity and associativity as axioms; I was not taught how to prove them using induction. But there is a pedagogical question regarding what sort of "easier" axiom-systems would be better than PA *to* start with. >In defense of your point I would say that I was next taught >commutativity and associativity as axioms; I was not taught how >to prove them using induction. >But there is a pedagogical question regarding what sort of "easier" >axiom-systems would be better than PA *to* start with. An induction schema that is restricted to *first-order* formulas only cannot possibly be pedagogically the right way to start. When I was taught induction in school, it was left vague what kinds of "properties" one can apply induction to. Concrete examples were given, all of which a logician could easily express as first-order formulas, but the question "what is a property?" was never directly addressed. Pedagogically, this has to be better than trying to specify precisely what properties are acceptable. Anyway, my main point was that your assertion that all people of a certain age "started with PA" is plainly false (or, in GeorgeGreenese, you LIED when you said that). SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences On Dec 19, 11:38 am, tc...@lsa.umich.edu wrote: > When I was taught induction in school, it was left vague what kinds of > "properties" one can apply induction to. Well, the whole question of when you should even start to teach the official definition of "what a first-order language is" does arise there. But I didn't personally know how to do anything more complex at the time, so in practice I personally was certainly limited to that. > Concrete examples were given, > all of which a logician could easily express as first-order formulas, Exactly. So it is NOT like YOU were doing something MORE complicated than that. Nor was I. > but the question "what is a property?" was never directly addressed. It wasn't even FORMULATED. BY definition AND PRACTICE, a property was whatEVER you KNEW HOW to WRITE down in a HOLE in which a PEG "of type property" would fit. This was very much a grammatical question. It was not (at this juncture) the philosophical question it would need to be later. > Pedagogically, this has to be better than trying to specify precisely > what properties are acceptable. This is completely misguided. You CANnot "specify" until AFTER you have some LARGE universe of "potential possible" properties out of which you could bless some and curse others. It is this universe which (at age 14, for most people) is NOT going to exis yet. What you DO have at that point is some conventions for writing things down. You have some known strings that are acceptable. You have some vague heuristics for gluing them together to make more complicated ones. THAT IS SUFFICIENT. That is NOT going to take anyone accidentally BEYOND any first-order considerations. So there is neither need NOR POSSIBILITY (at THAT juncture) of "specifying precisely what properties are acceptable". > Anyway, my main point was that your assertion that all people of a certain > age "started with PA" is plainly false (or, in GeorgeGreenese, you LIED > when you said that). No, really, it isn't. People did and do generally start with the identities and those ARE axioms of PA. As I have already said, what people really start with is calculation, is the whole 0th-order piece. The question is, what about the first- order piece? How does THAT arise/arrive? When does it become important to start saying things about all natural numbers beyond how to add 1 to them? Seriously, what is the first contentful first-order generalization about the naturals that you personally ever learned? I am thinking in my case that it was commutativity and associativity but I am pretty sure those were presented to me as axioms. As I said before, "starting WHAT?" -- my point being that at some point one starts to prove 1st-order sentences in arithmetic. Maybe in my case the first ones were about odd/even results -- the sum of two odd numbers or two even numbers is even, the sum of an odd and an even is odd; the product of odds is odd; the product of an even and anything else is even. Those all have fairly straightforward 1st-order proofs if you can use associativity. You don't actually need induction for them. In any case, none of that is the point. The point IS that ALL the axioms of PA EXCEPT induction involve/embody simple straightforward truths about the naturals that most people WERE taught early on and WERE using THE WHOLE time. And your point about what to "specify" for induction is just bullshit. When induction is introduced, the simple introductory context into which it is being introduced will guarantee that nothing beyond 1st-order comes up. >And your point about what to "specify" for induction is just bullshit. >When induction is introduced, the simple introductory context into >which it is being introduced will guarantee that nothing beyond 1st-order >comes up. Even granting a lot of things in your article that I don't agree with, this *still* makes no sense. Why PA? Why not PRA, or even weaker induction schemes? These would all suffice for anything that comes up in high school. For that matter, why PA and not some two-sorted first-order system like RCA_0 or ACA_0? They would all suffice too. In fact, in many ways they're much more natural than PA. That everyone of a certain age "started with PA" is simply a lie. SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences On Dec 19, 11:38 am, tc...@lsa.umich.edu wrote: > An induction schema that is restricted to *first-order* formulas only cannot > possibly be pedagogically the right way to start. This is completely missing the point. The issue is not how to phrase the induction schema. The prior issue is WHAT KIND OF LANGUAGE is math IN GENERAL being conducted in, in your classroom? If the students are used to writing 1st-order generalizations then it will automatically occur to them to be putting that kind of peg in the hole. If they are not then it gets more complicated. But my point is, it NEVER gets 2nd-order complicated, or complicated in a way that could cause theoretical problems, because the students HAVEN'T LEARNED THAT yet. My point is that your WHOLE DISCOURSE was restricted to *0th*-order formulas and NATURAL-language generalizations for far too long. Getting in the habit of writing things with quantifiers is almost un- natural. One DOES NOT SAY (or write) Ex[2*x=y]. One RATHER says, "y is even". One might even say, "since y is even, let x=y/2". This does NOT APPEAR to be invoking 1st-order ANYthing. My point is, you don't have to be thinking about phrasing it in the canonical first-order grammar TO BE PERFORMING first-order reasoning IN PA. > George, you've *got* to be kidding here. Do you not have any clue how > idiosyncratic your viewpoint is? George knows very well by now that his views on many issues are quite bizarre and idiosyncratic. He's just convinced every one else is wrong. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus >> George, you've *got* to be kidding here. Do you not have any clue how >> idiosyncratic your viewpoint is? > >George knows very well by now that his views on many issues are quite >bizarre and idiosyncratic. He's just convinced every one else is wrong. I know that much, but here he went further, by making claims about "almost everyone ten years younger than...," i.e., attributing his idiosyncratic views to a large population of other people. SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences On Dec 19, 7:25 am, Aatu Koskensilta <aatu.koskensi...@xortec.fi> wrote: > George knows very well by now that his views on many issues are quite > bizarre hardly. > and idiosyncratic. Well, unpopular, yes. > He's just convinced every one else is wrong. No, really, I'm not. The reason why I am always in caps with harsh language is that the positions I am attacking are just blatantly obviously incoherent. The people who are asserting them don't, for the most part, believe them themselves. NObody believes them. They aren't coherenty believABLE. Prof.Smith and I have been talking, for example, about the intended model vs. the formal language. He is the one who said that he didn't think formal languages should even be referred to as a language. That is considerably less defensible than anything *I* have ever said. It really is just plain obvious that anyone can posit a language and some axioms and inquire about what models of the axioms might exist. Insisting that mathematicians "usually" have an intended model to begin with instead IS SILLY. WHERE DID THEY GET this model from? HOW did they DEFINE it?? My point is simply that there were axioms involved THERE, TOO. They can't escape. It's not my way or the highway: there IS NO highway. My way IS THE way. Even the people who are claiming NOT to be on it OBVIOUSLY ARE. > The reason why I am always in caps with harsh language > is that the positions I am attacking are just blatantly obviously > incoherent. The people who are asserting them don't, for the > most part, believe them themselves. NObody believes them. > They aren't coherenty believABLE. Ah, it's all explained now: you do not in fact believe the bizarre assertions you habitually sputter. You know 'tis so, deep down your heart. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus >> George knows very well by now that his views on many issues are >> quite bizarre >> > hardly. But they are - sometimes, george! >> and idiosyncratic. >> > Well, unpopular, yes. Well, not generally accepted, right... ;-) > Prof. Smith and I have been talking, for example, about the > intended model vs. the formal language. He is the one who > said that he didn't think formal languages should even be referred > to as a language. That is considerably less defensible than anything > *I* have ever said. Well..., Alonzo Church [you know that guy?] also mentioned such a view, in one of his papers. He writes: "We distinguish between a /logistic system/ and a /formalized language/ on the basis that the former is an abstractly formulated calculus for which no interpretation is fixed, and thus has a syntax and no semantics; but the latter is a logistic system together with an assignment of meanings to its expressions. [...] In order to obtain a formalized language it is necessary to add to these /syntactical rules/ of the logistic system, /semantical rules/ assigning meanings (in some sense) to the well-formed expressions of the system." (Alonzo Church, The Need for Abstract Entities, 1951) [ Oh right, this is an ancient text and etc. etc. *sigh* ] > Insisting that mathematicians "usually" have an intended model to > begin with instead IS SILLY. Nonsense. (Clearly historically most mathematical "theories" started out _without_ being formalized as axiomatic systems. At least this is true for _set theory_, as you certainly will know.) > WHERE DID THEY GET this model from? Mathematical intuition? F. SignatureE-mail: info<at>simple-line<dot>de > Well..., Alonzo Church [you know that guy?] also mentioned such a view, > in one of his papers. He writes: [quoted text clipped - 11 lines] > > (Alonzo Church, The Need for Abstract Entities, 1951) Exactly the distinction that well-brought-up logicians still make (if not necessarily using those words) :-))) > > Well..., Alonzo Church [you know that guy?] also mentioned such a view, He mentioned it in 1951. This was well before people were in the habit of talking about regular languages, context-free languages, or anything else in a hierarchy of PURELY formal languages. NObody, NOT EVEN Alonzo Church, NOWadays, gets to try to put THAT genie back in the bottle. There simply now IS a restricted sense of "formal language" in which certain sets of strings are formal languages. > > In order to obtain a formalized language Well, certainly, if you start out with Peter Smith's sense of "language" and then try to make THAT "more" formal, then, yes, > > it is necessary to add to > > these /syntactical rules/ of the logistic system, /semantical rules/ > > assigning meanings (in some sense) to the well-formed expressions of the > > system." > > (Alonzo Church, The Need for Abstract Entities, 1951) If on the other hand you just start with a purely formal language to begin with, the question of a "formalized language" is simply moot. The language didn't NEED to be formalIZED because it was already totally formal. What it needed to be, according all three of Fritsche, Smith, and Church, here, was semanticized or interpreted. > Exactly the distinction that well-brought-up logicians still make (if > not necessarily using those words) :-))) This is not a distinction and you are lapsing into lying and ignorance. What assigns meanings to the wffs of the system is DESIGNATING SOME OF THEM AS AXIOMS. This is an act that does have semantic content simply because axioms have to be true, but that is not the point. The point is that the axioms are characterizABLE syntactically AS OPPOSED to semantically and that they confer NOT semantic truth BUT RATHER syntactic provability. As a result of which they do NOT need to be interpreted in order to communicate. > What assigns meanings to the wffs of the system is DESIGNATING > SOME OF THEM AS AXIOMS. Really? Well that sounds like magic to me. If I tell you that "mae glo yn du" is true in Welsh [heck, hope I've remembered that right], you don't thereby get to know what it means. Telling you the same about some other Welsh sentences won't help either. Telling you that in my fave axiomatized system "Fa" is an axiom (or is an axiom and true), you don't thereby get to know what it means. Telling you the same about some other sentences of the system won't help either. > > What assigns meanings to the wffs of the system is DESIGNATING > > SOME OF THEM AS AXIOMS. > > Really? Well that sounds like magic to me. So what assigns meaning to the wffs? > If I tell you that "mae glo yn du" is true in Welsh [heck, hope I've > remembered that right], you don't thereby get to know what it means. [quoted text clipped - 5 lines] > Telling you the same about some other sentences of the system won't > help either. > > What assigns meanings to the wffs of the system is DESIGNATING > > SOME OF THEM AS AXIOMS. > Really? Well that sounds like magic to me. Oh, it is. You should be more than impressed. More seriously, though, obviously there is more than the designation going on. These are FIRST-ORDER formal languages we are talking about and the rules of first-order grammar and the rules OF INFERENCE IN FIRST-ORDER LOGIC are also necessary to cause axiomatizations to have meanings. But my point is, ALL OF THOSE ARE CHARACTERIZABLE SYNTACTICALLY, so, again, SEMANTIC INTERPRETATION is NOT necessary. Something LIKE "interpretation" is necessary for the LOGICAL part of the vocabulary, perhaps, but again, that is performable syntactically. > If I tell you that "mae glo yn du" is true in Welsh [heck, hope I've > remembered that right], you don't thereby get to know what it means. THIS is NOT like THAT! > Telling you the same about some > other Welsh sentences won't help > either. If, however, some of the words were logical connectives and you DID manage to figure out those, and if you had a specification of ALL the provable things in the language, then help might eventually occur, or at least insight. For you to be trying this with a finite number of assertions about the concrete world, when you know that OUR subject matter deals with an infinite number of axioms about abstractions, is, well, I hope it is just a way of designing an introductory lecture on the topic. It is not serious. The natural-language sentences you are whining about will fail here because THOSE SENTENCES ARE ABOUT THE REAL WORLD. But if they are about abstractions or math then it is ENTIRELY different. If pattern is all they are TRYING to communicate IN THE FIRST PLACE, then this can and does work, simply because there is nothing beyond the patterning FOR the symbols TO mean, in the abstract context. > Telling you that in my fave axiomatized system "Fa" is an axiom (or is > an axiom and true), you don't thereby get to know what it means. It's an axiom SYSTEM. This axiom won't occur by itself. > Telling you the same about some other sentences of the system won't > help either. I repeat, set theory, PA, and every other 1st-order theory you know IS A FACTUAL COUNTEREXAMPLE. WhatEVER "epsilon" means, it means in virtue of the axioms of ZFC AND NOTHING ELSE (except maybe of the definition of FOL). Of course, that theory is interpretable into many different structures, but what will you THEN allege? That before the language is interpreted, it means NOTHING? Here is how stupid you sound: You, the distinguished Prof.Peter Smith, are saying the following: 2+5=7 doesn't mean anything. It doesn't say anything about anything. Now, if it were interpreted, if I said 2 cars + 5 cars = 7 cars, or starting with 2 cups of sugar and adding 5 cups of sugar yields 7 cups of sugar, THEN THAT would be saying something. But both of those would be INTERPRETING 2+5=7. UNTIL you do something like that, 2+5=7 can't mean anything. The actual truth of the matter is that it is precisely BECAUSE 2+5=7 DOES mean something by itself (and something we can prove from axioms, at that) that WE CAN KNOW that 2 eggs plus 5 eggs will be 7 eggs, AND ALL THOSE OTHER consequences. The uninterpreted version is actually MORE important. Of course, on your side of the fence, this state of affairs is so intolerable that you will make up abstract entities for 2, 5, and 7 to actually "be" so that you CAN interpret them. > Here is how stupid you sound: > You, the distinguished Prof.Peter Smith, are saying the following: > 2+5=7 doesn't mean anything. Am I? That's odd. I always thought that "2 + 5 = 7" means that two plus five is seven ... but obviously I must have got that wrong. > > You, the distinguished Prof.Peter Smith, are saying the following: > > 2+5=7 doesn't mean anything. > Am I? That's odd. I always thought that "2 + 5 = 7" means that two > plus five is seven ... but obviously I must have got that wrong. Well, you got the quotes wrong, for starters. I was talking about your opinion about the meaning of 2+5=7. You responded with your opinion about the meaning of "2+5=7". Obviously if we are going to interpret all 3 of "2", 2, and two, then we are not going to run out of levels of interpretation, at least not quickly. The argument was ABOUT what things NEED to be interpreted, and whether. Are you even psychologically capable of sticking to the subject? The one thing that makes the quotes acceptable is that strings are known to be abstract. But if you are going to wax realist about the two that 2 refers to, then 2 and two are abstract as well. > > > You, the distinguished Prof.Peter Smith, are saying the following: > > > 2+5=7 doesn't mean anything. [quoted text clipped - 4 lines] > Well, you got the quotes wrong, for starters. > I was talking about your opinion about the meaning of 2+5=7. Got quotes wrong too, have I? Gosh. And there I was thinking that I was conforming to normal good practice with use and mention. Ah well, ... > If I tell you that "mae glo yn du" is true in Welsh [heck, hope I've > remembered that right], you don't thereby get to know what it means. > Telling you the same about some other Welsh sentences won't help > either. There are logical symbols as well. They provide a sort of skeleton for the structure. In particular, IF Welsh has a word for "not", then it arguably IS possible, JUST from surveying the true sentences, to discover it. > What assigns meanings to the wffs of the system is DESIGNATING > SOME OF THEM AS AXIOMS. This is an act that does have semantic > content simply because axioms have to be true [...]. Wait a second. Aren't those axioms /true/ only when /interpreted/ (i.e. in a model)? Hence isn't it the /interpretation/ that assigns truth to an axiom? F. SignatureE-mail: info<at>simple-line<dot>de > > What assigns meanings to the wffs of the system is DESIGNATING > > SOME OF THEM AS AXIOMS. This is an act that does have semantic [quoted text clipped - 3 lines] > (i.e. in a model)? Hence isn't it the /interpretation/ that assigns > truth to an axiom? Well, they're true in all their models, by definition. Last time George and I spoke on this point, he was of the opinion, if I understood him correctly, that interpreting the axioms in structures which falsify (one or more of) them doesn't count -- axioms are always taken as true regardless of how they are interpreted; interpretation is otiose. I'll admit it seems like a rather pointless exercise to interpret axioms in structures in which they are false, it just happens to be involved in the Tarskian conception of logical consequence: in those structures too the axioms imply their theorems /and nothing else/. -- hz > [herb thinks George thinks] interpreting the axioms in > structures which falsify (one or more of) them doesn't count -- > axioms are always taken as true regardless of how they are > interpreted; interpretation is perhaps I should say "supererogatory"! :-) -- hz > > Hence isn't it the /interpretation/ that assigns > > truth to an axiom? This is what is known as "a stupid question". Maybe FF should look in the dictionary under "Axiom". Axioms have to be provable because THAT'S THE DEFINITION OF "Axiom". The Axioms are the things that you just presume true FROM THE BEGINNING, the things to which you PRE-assign a definitional proof-length of 0 (or of 1 if takes some effort to show that some string is recognizable as an axiom). > Well, they're true in all their models, by definition. But axioms are NOT UNIQUE in that regard! That is NOT a special property Of Axioms! EVERY last wff is true in all of ITS models! MODELS have to be models OF something! It is, as I was trying to beat through MoeBlee's head a minute ago, "structures" and "interpretations" that get to be unattached. If a structure is a model of a wff then the the wff is true in the structure. MoeBlee's point was that people were in a very generalized habit of using "model" withOUT requiring it to be a model of anything in PARTICULAR. That is an observable fact about the brute weight of usage; neither I nor anyone else GETS to disagree with it, so MoeBlee probably thought it would therefore be safe, or at least defensible, to assert it. He was entirely wrong about that. People's general habit of doing this IS BAD. It is sloppy. IT NEEDS reform. People NEED to clean up their (usage)ACT. Instead, MoeBlee chose to DEFEND the fact that people generically tend to talk this way AS ACCEPTABLE because it is allowed by definitions that he can quote from Enderton. That is using a good work to confuse people and make discourse in general LESS accurate and it is not intellectually acceptable behavior. > Last time George and I spoke on this point, he was of the opinion, > if I understood him correctly, that interpreting the axioms in > structures which falsify (one or more of) them doesn't count -- > axioms are always taken as true regardless of how they are > interpreted; interpretation is otiose. Close, yeah. It amazes me that some people can just tune into my wavelength (if they feel like it) while others must insist that I'm just evil. > I'll admit it seems like a rather pointless exercise to interpret > axioms in structures in which they are false, it just happens > to be involved in the Tarskian conception of logical consequence: > in those structures too the axioms imply their theorems /and > nothing else/. The verb "imply" is the key word in that sentence. Material implication is convenient in some ways and misleading in others; please let us NOT have another thread about "vacuously true". >> Hence isn't it the /interpretation/ that assigns >> truth to an axiom? >> > Maybe FF should look in the dictionary under "Axiom". No, really not. > Axioms have to be provable because THAT'S THE DEFINITION OF "Axiom". Right. > The Axioms are the things that you just presume true FROM THE > BEGINNING, ... Oh, wait a second. Aren't you mixing up TRUTH with PROVABILITY here? :-o F. SignatureE-mail: info<at>simple-line<dot>de > > The Axioms are the things that you just presume true FROM THE > > BEGINNING, ... > > Oh, wait a second. Aren't you mixing up TRUTH with PROVABILITY here? :-o It's not MIXING if they never got SEPARATED TO BEGIN with! It is if you DO have structures/interpretations/semantics that the concepts are SEPARATE and you have to draw a distinction. If there simply ARE No models or interpretations at all then the ONLY way ANYthing ever gets to be true is BY being proved. (P v ~P) is tautological and therefore axiomatic and therefore true, but withOUT models, what can you say about the truth of P if it is not a theorem, EVEN though you know (P v ~P) ? NOTHING, THAT'S what. My point is that we are obviously in two different worlds here. One is the usual world where semantics is relevant, and the other is the post-completeness-theorem world where one has THE OPTION of simply ignoring first-order semantics altogether, SINCE THE COMPLETENESS THEOREM EXPLAINS how to re- do that SYNTACTICALLY. It MATTERS WHETHER "it's true" VERSUS "it's provable" IN THE WORLD THAT HAS semantics. Since I am inviting people into the world THAT DOESN'T have semantics, that distinction is NOT stressable versus ME! george wrote: > > > Hence isn't it the /interpretation/ that assigns > > > truth to an axiom? [...] > > Well, they're true in all their models, by definition. > > But axioms are NOT UNIQUE in that regard! > That is NOT a special property Of Axioms! > EVERY last wff is true in all of ITS models! All true, of course. > MODELS have to be models OF something! > It is, as I was trying to beat through MoeBlee's head a minute ago, > "structures" and "interpretations" that get to be unattached. MoeBlee's assertion that it is a theorem that every structure is a model of some theory is a new thought to me. Even if it is so, it does seem to me to be ill-advised to use "structure" and "model" interchangably (although I'm as slack as anyone else on using "structure" and "interpretation" more or less interchangeably). > If a structure is a model of a wff then the the wff is true in the > structure. Of course. > MoeBlee's point was that people were in a very generalized habit of > using [quoted text clipped - 16 lines] > intellectually > acceptable behavior. I agree that it's a bad habit that needs to be reformed, and I'm willing to do my bit to be the annoying guy who keeps pointing out the misuse. Assuming MoeBlee is correct in asserting that Enderton's definitions allow calling every structure a model, I would guess that that sort of usage would vary by context -- one would have to take some care not use one term for the other indiscriminately. > > Last time George and I spoke on this point, he was of the opinion, > > if I understood him correctly, that interpreting the axioms in [quoted text clipped - 6 lines] > wavelength (if they feel like it) while others must insist > that I'm just evil. Like anyone else, when I'm in an argument I'm inclined to reject EVERYTHING my opponent says, no matter how innocuously and obviously true some of it may be. This is so obviously a form of ad hominem (If this jerk says X, then X must be false) that it's particularly embarassing, as a psuedo-logician, to fall prey to it. It's rhetorically bad, too, to be caught denying what's plainly true. > > I'll admit it seems like a rather pointless exercise to interpret > > axioms in structures in which they are false, it just happens [quoted text clipped - 3 lines] > > The verb "imply" is the key word in that sentence. Yes, and that's _my_ little hobby-horse. > Material implication is convenient in some ways and misleading in > others; please let us NOT have another thread about "vacuously true". That's not my intent, nor am I asserting that axioms (or any other sentence) "vacuously imply" their consequences when interpreted in structures in which they (the axioms or sentences) are false (with the unfortunate exception of contradictions -- which "vacuously imply" every sentence, regardless of the structure in which they are interpreted, according to the standard (i.e. Tarskian) definition of logical consequence). -- hz > > MoeBlee's assertion that it is a theorem that every structure is a > model of some theory is a new thought to me. Even if it is so, > it does seem to me to be ill-advised to use "structure" and "model" > interchangably (although I'm as slack as anyone else on using > "structure" and "interpretation" more or less interchangeably). While at the issue of structure/interpretation/model let me point a couple of related points, which is basically that, in any one of the 3 just mentioned ,a degree of subjectivity (relativity) must necessarily exist. For instance, given the language L(a,b,<) we'd have 2 theories say T1 = {a < b} and T2 = {~(a < b)}. Now given a particular ZF ordered pair say m = (a',b'). The point is in and of itself, the *set* m is neither an model nor an interpretation (nor a "structure", depending on one's definition of the word). The missing link here is that the mental (hence subjective/relative) mapping M between m and one of the T1 or T2 must necessarily exist. "Necessarily" here is because in strict Hilbert's syntactical formalism, there are just [syntactical] axioms, rules of inference, and (syntactical) provability. So literally we couldn't prove any undecidability by by mentioning the word "true" or "false", because there is none of such word in this Hilbert's strictly syntactical paradigm. In a nutshell, Godel introduced an addendum to this strict paradigm by introducing the _subjectivity_ of "arithmetic truths". (We could discuss this if that's desired). In any rate, sometimes in the past, I mentioned to MoeBlee and others that if one *loosely* considers the set m above as a "structure" then one could see a model of a theory is nothing but a mere _subjective_ mapping M between m and a *chosen*theory. So when *one* talks about a particular model, one should be prepared for the fact that others just might have another *different* mapping M' between m and whatever the other theory be. So in a slack usage, *with some appropriate explanation*, there's nothing wrong to say when one considers a model for a theory, others might not consider it as a model of the same theory (subjectively). Moeblee seemed to be bogged-down with a lot of terminology-technicalities to see the "bigger picture" of this model subjectivity. > -- > hz > In any rate, sometimes in the past, I mentioned to MoeBlee and others > that if one *loosely* considers the set m above as a "structure" then > one could see a model of a theory is nothing but a mere _subjective_ mapping > M between m and a *chosen*theory. Yeah, well, I DON'T loosely take 'structure' defined as you do. I use the technical definition given by a certain text (or, if other people use a different formulation from another text, then I'm happy to use the other formulation while I note the way in which it is just a different technical way to achieve a formalization of the same notion). > Moeblee seemed > to be bogged-down with a lot of terminology-technicalities to see > the "bigger picture" of this model subjectivity. Your 'bigger picture' is a mural of confusion. And my being careful as to technical definitions does not disallow me from seeing a coherent bigger picture. Moeblee > MoeBlee's assertion that it is a theorem that every structure is a > model of some theory is a new thought to me. What is difficult about it? (1) By definition, M is model of a set of sentences G iff M is a structure for the language of G and every member of G is true in M. So every model is a structure, by definition. (By the way, in another post I think I might have allowed G to be a set of formulas. But I think G should be a set of sentences.) (2) Every structure M is a model of the set of valid sentences in the language that M is a structure for. So every structure is a model. > Even if it is so, > it does seem to me to be ill-advised to use "structure" and "model" > interchangably (although I'm as slack as anyone else on using > "structure" and "interpretation" more or less interchangeably). In such contexts as I mentioned, what is the harm of using 'model' and 'structure' interchangably, especially the context I mentioned?: (1) M is a model for a language (2) M is model of a set of sentences (3) M is structure for a language. (4) M is structure of a set of sentences. Especially, when the precise definitions are given. (2) and (3) are more common than (1) and (4), though (1) is found in, for example, Chang & Keisler, though, (4) is admittedly rather odd sounding and therefore I don't use it, but, as long as my definitions have been clearly stipulated, it wouldn't be harmful if I did use (4) even though I don't prefer it. > > It amazes me that some people can just tune into my > > wavelength (if they feel like it) while others must insist [quoted text clipped - 6 lines] > embarassing, as a psuedo-logician, to fall prey to it. It's rhetorically > bad, too, to be caught denying what's plainly true. Except we don't have an example of anyone disagreeing with George simply because he is otherwise a royal jerk. MoeBlee > > MoeBlee's assertion that it is a theorem that every structure is a > > model of some theory is a new thought to me. [quoted text clipped - 4 lines] > structure for the language of G and every member of G is true in M. So > every model is a structure, by definition. Right. > (By the way, in another post I think I might have allowed G to be a > set of formulas. But I think G should be a set of sentences.) OK. > (2) Every structure M is a model of the set of valid sentences in the > language that M is a structure for. Yup. > So every structure is a model. Of any validity, yes. Can we say further that every structure is a model of a theory (other than the null theory containing only validities)? Shall we agree as to what sets of sentences constitute "a theory" first? I usually think of "a theory" as having a recursive (or at least r.e.) set of axioms. And as being consistent (i.e. having a model)! > > Even if it is so, > > it does seem to me to be ill-advised to use "structure" and "model" [quoted text clipped - 3 lines] > In such contexts as I mentioned, what is the harm of using 'model' and > 'structure' interchangably, especially the context I mentioned?: If I recall correctly, this started when you said to Nam: > > Okay, in a technical sense, '2=1+1' is true relative to models because > > it's true in some models but not in others. To which George objected: > No, true in some interpretations but not in others. which might seem like a rather pedantic distinction (or, as you seem to think, a false distinction), but I think that in this area with Nam you have to be very precise, and the distinction is merited. > (1) M is a model for a language > [quoted text clipped - 11 lines] > have been clearly stipulated, it wouldn't be harmful if I did use (4) > even though I don't prefer it. Sure, as long as you clearly stipulate your definitions, no problem. I think the default usages are, as you point out, (2) and (3). > > > It amazes me that some people can just tune into my > > > wavelength (if they feel like it) while others must insist [quoted text clipped - 9 lines] > Except we don't have an example of anyone disagreeing with George > simply because he is otherwise a royal jerk. On the contrary, I think that people in this forum tend to be argumentative when they are contradicted. They often will not take a brusque correction with equanimity. They will put uncharitable and even unreasonable constructions on what has been said to them. They will move heaven and earth to show that they were not, in fact, wrong. Do you want documentation? I'd prefer not to name names. Also, that would be a very tedious chore. I'd like to take this opportunity to say that George has never been a jerk to me. Of course I attribute this to his acute perception of my sterling character, but it's more probably that I'm not edjicated enough to merit abuse. I hope one day to be smart enough to rate an "oh, SHUT UP" from George. In general I find his explanations of things to be quite patient and lucid. Of course, he is occasionally wrong. So what? -- hz > > > MoeBlee's assertion that it is a theorem that every structure is a > > > model of some theory is a new thought to me. [quoted text clipped - 21 lines] > Of any validity, yes. Can we say further that every structure is a > model of a theory (other than the null theory containing only validities)? I'm not familiar with the expression 'null theory' to refer to the set of validities, but I'll go along with it. As to the question, I'd have to think about it. > Shall we agree as to what sets of sentences constitute "a theory" first? For simplicity, let's confine to classical first order. Authors such as Enderton take a theory to be any set of sentences closed under entailment (which, thanks to the completeness theorem, is a set of sentences closed under provability). Authors such as Chang & Keisler take a theory to be any set of sentences. And some authors take a theory to be a pair <S E> where S is a set of sentences and E is the set of sentences entailed by S (which, thanks to the completeness theorem, is the set of theorems of S). I adopt Enderton's definition. > I usually think of "a theory" as having a recursive (or at least r.e.) > set of axioms. That is a recursively axiomatized theory. There are theories that are not recursively axiomatized. > And as being consistent (i.e. having a model)! Those are consistent theories. There are theories that are not consistent. > > > Even if it is so, > > > it does seem to me to be ill-advised to use "structure" and "model" [quoted text clipped - 16 lines] > to think, a false distinction), but I think that in this area with > Nam you have to be very precise, and the distinction is merited. Anyone may state definitions and explicate a discussion on the basis of those definitions. Meanwhile, what I said is precisely correct given ordinary defintions: '2=1+1' is true in some models and not true in other models. That is precisely correct. What is cloudy is the terminology 'true relative to models', which is Nam's terminology. I don't use that terminology. All I said (or meant to convey) in the passage you mentioned is that I could see a sense in which that terminology could be understood. So, again, to be clear: My terminology is 'true in a model' and it is precisely correct that '2=1+1' is true in some models and not true in other models. Nam's terminology is 'true relative to a model', and though I do not in any way claim to arbitrate what HE means by that, my point was just to say that IF he means 'true in a model', then yes, of course, '2=1+1' is true in some models and not true in other models. Then, there were the rest of my remarks. > > (1) M is a model for a language > [quoted text clipped - 14 lines] > Sure, as long as you clearly stipulate your definitions, no problem. > I think the default usages are, as you point out, (2) and (3). More common, not necessarily default. > > > > It amazes me that some people can just tune into my > > > > wavelength (if they feel like it) while others must insist [quoted text clipped - 17 lines] > documentation? I'd prefer not to name names. Also, that would be > a very tedious chore. I'm not interested in such tedium. But to be convinced I would have to know what examples you have in mind specifically regarding George, since I don't know of an instance of someone disagreeing with George merely for his being a jerk. > I'd like to take this opportunity to say that George has never been > a jerk to me. Of course I attribute this to his acute perception of > my sterling character, but it's more probably that I'm not edjicated > enough to merit abuse. I hope one day to be smart enough to rate > an "oh, SHUT UP" from George. In general I find his explanations > of things to be quite patient and lucid. I find his explanations usually to be bizarre. > Of course, he is occasionally wrong. So what? I think he's more than occasionally wrong, and worse, he's a jerk while being wrong. MoeBlee > > > > MoeBlee's assertion that it is a theorem that every structure is a > > > > model of some theory is a new thought to me. [quoted text clipped - 24 lines] > I'm not familiar with the expression 'null theory' to refer to the set > of validities, but I'll go along with it. I probably just made it up, although it seems right. It's the theory with no non-logical axioms (ie only validities as theorems). It seems to me that this is a degenerate case of the word "theory": usually I'd expect a "theory" (as opposed to a "logic") to _assume_ a logic and to have some non-logical axioms/theorems. > As to the question, I'd have to think about it. Bingo. That's what's "a new thought to me": that an arbitrary structure is a model of some non-null theory. I don't happen to know whether that's true. > > Shall we agree as to what sets of sentences constitute "a theory" first? > [quoted text clipped - 9 lines] > > I adopt Enderton's definition. OK. > > I usually think of "a theory" as having a recursive (or at least r.e.) > > set of axioms. > > That is a recursively axiomatized theory. There are theories that are > not recursively axiomatized. Yes, they seem a little imaginary to me, but I'm willing to go along with it. > > And as being consistent (i.e. having a model)! > > Those are consistent theories. There are theories that are not > consistent. It would seem that there is but one such theory over any signature: the theory that consists of all sentences in the language. This seems like another degenerate case of "theory". I'm willing to accept the informal locution "inconsistent theory"; I'm willing to accept an "inconsistent theory" as existing under the Enderton definition of "theory". But the extremal cases of "theories" that contain only validities, or that contain every sentence, do seem a little silly to me. I guess it's the price you pay for definitional elegance. > > > > Even if it is so, > > > > it does seem to me to be ill-advised to use "structure" and "model" [quoted text clipped - 39 lines] > true in some models and not true in other models. Then, there were the > rest of my remarks. Yuh, I guess the argument turns on whether you want to affirm a) 2 = 1 + 1 is false in no model (of PA) or b) 2 = 1 + 1 is false in some models (of the language of PA). It seems to be a simple ambiguity to be resolved. I personally would prefer (a) and would assume that you are using "model" in the sense of (a) in the absence of an explicit definition otherwise. I concede that it is possible that Nam uses the word "model" in some sense other than (a). I acccept that the literature allows the usage of "model" in the sense of both (a) and (b). But, as I said before, this seems ill- advised, in that it allows this ambiguity to arise. > > > (1) M is a model for a language > > [quoted text clipped - 16 lines] > > More common, not necessarily default. Well, time will tell, perhaps. > > > > > It amazes me that some people can just tune into my > > > > > wavelength (if they feel like it) while others must insist [quoted text clipped - 22 lines] > since I don't know of an instance of someone disagreeing with George > merely for his being a jerk. Such instances would, of course, require an inference as to someone's motives, since no one's going to assert that they are disagreeing just to be contrary. That will remain, irredeemably, a matter of opinion. I'd rather just state my opinions and leave it at that. > > I'd like to take this opportunity to say that George has never been > > a jerk to me. Of course I attribute this to his acute perception of [quoted text clipped - 4 lines] > > I find his explanations usually to be bizarre. Not usually, but sometimes. Usually there is a core of something insightful even in the seemingly bizarre assertions. > > Of course, he is occasionally wrong. So what? > > I think he's more than occasionally wrong, and worse, he's a jerk > while being wrong. Actually, I find it even more annoying when someone is being a jerk while being right! -- hz > > I'm not familiar with the expression 'null theory' to refer to the set > > of validities, but I'll go along with it. [quoted text clipped - 4 lines] > I'd expect a "theory" (as opposed to a "logic") to _assume_ a logic > and to have some non-logical axioms/theorems. You may consider it a degenerate or trivial case, but it is still a theory by such ordinary defiinitions of 'theory' that I mentioned. > > As to the question, I'd have to think about it. > > Bingo. That's what's "a new thought to me": that an arbitrary structure > is a model of some non-null theory. I don't happen to know whether > that's true. I'd have to clear some technicalities, but I suspect it is true. But in any case, every structure is a model of the set of valid sentences (in the language that the structure is a structure for). And that set of valid sentences is a theory, even if only trivially so. Therefore, every structure is a model of some theory, even if only of the theory that is the set of valid sentences in the language that the structure is a structure for. > > > Shall we agree as to what sets of sentences constitute "a theory" first? > [quoted text clipped - 20 lines] > Yes, they seem a little imaginary to me, but I'm willing to go along > with it. They are famous and important. Most famous in that way is perhaps the theory that is the set of sentences true in the standard model for the language of PA. It is a famous theorem that that theory is not recursively axiomatized (or 'not recursively axiomatizable' if you prefer). That there are theories that are not recursively axiomatized (or 'not recursively axiomatizable' if you prefer) is a famous and important subject in mathematical logic. > > > And as being consistent (i.e. having a model)! > [quoted text clipped - 3 lines] > It would seem that there is but one such theory over any signature: > the theory that consists of all sentences in the language. Right. > This > seems like another degenerate case of "theory". If you prefer to think of it that way. > I'm willing to > accept the informal locution "inconsistent theory"; I'm willing [quoted text clipped - 4 lines] > or that contain every sentence, do seem a little silly to me. I > guess it's the price you pay for definitional elegance. Whether it seems silly, it still is indeed the case, from the definitions. > > My terminology is 'true in a model' and it is precisely correct that > > '2=1+1' is true in some models and not true in other models. Nam's [quoted text clipped - 11 lines] > > b) 2 = 1 + 1 is false in some models (of the language of PA). I affirm both. But the particular remark I made didn't mention PA. > It seems to be a simple ambiguity to be resolved. I personally would > prefer (a) and would assume that you are using "model" in the sense > of (a) in the absence of an explicit definition otherwise. I concede > that it is possible that Nam uses the word "model" in some sense > other than (a). There is no conflict among a), b), and my use of the word 'model'. As to Nam's nottions, that's a whole other matter. > I acccept that the literature allows the usage of "model" in the > sense of both (a) and (b). But, as I said before, this seems ill- > advised, in that it allows this ambiguity to arise. There is not an ambiguity in what I said. '2 = 1 + 1' is true in some models and false in other models Is easily formalized as: EBC(B is a model & C is a model & '2 = 1 + 1' is true in B & '2 = 1 + 1' is false in C) with B is a model <-> EG(G is a set of sentences & B is a model of G) And even without such pedantic formulations, it's just common sense in mathematical logic what is meant by: B is a model '1+1=2' is true in B. '1+1=2' is false in B. Nothing more than quite ordinary meanings in mathematical logic are needed to understand that '1+1=2' is true in some models and false in others. I could have said it equivalently as '1+1=2' is a contingent sentence. This is utterly straightforward stuff. > > > On the contrary, I think that people in this forum tend to be argumentative > > > when they are contradicted. They often will not take a brusque correction [quoted text clipped - 13 lines] > to be contrary. That will remain, irredeemably, a matter of opinion. > I'd rather just state my opinions and leave it at that. And I know of no evidence that anyone has disagreed with George on some point or another merely because he is a jerk. > > > I'd like to take this opportunity to say that George has never been > > > a jerk to me. Of course I attribute this to his acute perception of [quoted text clipped - 7 lines] > Not usually, but sometimes. Usually there is a core of something > insightful even in the seemingly bizarre assertions. I find it usual. Often, at least. And occasionally there is some bit of insight mixed in with his bizarre declamations, though hardly enough to justify. Also, of course, many of his statements are not bizarre and are good points. But I find that he's usually, or at least much much too often bizarre, especially with his continual out-of-the- blue decrees as to how things must be formulated, defined, regarded, or conceived to be correct. > > > Of course, he is occasionally wrong. So what? > [quoted text clipped - 3 lines] > Actually, I find it even more annoying when someone is being a jerk > while being right! I guess that's a joke or something. Of course, it's subjective what annoys a person. But it does seem disconnected to me to be more annoyed by someone who is at least correct. MoeBlee > > > I'm not familiar with the expression 'null theory' to refer to the set > > > of validities, but I'll go along with it. [quoted text clipped - 7 lines] > You may consider it a degenerate or trivial case, but it is still a > theory by such ordinary defiinitions of 'theory' that I mentioned. Indeed it is. > > > As to the question, I'd have to think about it. > > [quoted text clipped - 9 lines] > that is the set of valid sentences in the language that the structure > is a structure for. For any structure S in a language L the structure assigns a truth-value to every sentence of L. For every sentence phi of L, one of phi and ~phi will be true and one will be false in S. On the assumption that not all sentences of L are validities and their negations, then S will be a model of some sentences that are not validities, and hence will be a model of some non-null theory of language L. Perhaps you would like to clean that up a bit. The further question would be whether every structure is a model of some non-null recursively axiomatized theory. > > > > Shall we agree as to what sets of sentences constitute "a theory" first? > > [quoted text clipped - 9 lines] > > > > > I adopt Enderton's definition. \> > OK. > > > > I usually think of "a theory" as having a recursive (or at least r.e.) > > > > set of axioms. [quoted text clipped - 12 lines] > (or 'not recursively axiomatizable' if you prefer) is a famous and > important subject in mathematical logic. Yes, I've heard of that theory. It still seems a little imaginary to me, not being recursively axiomatizable. > > > > And as being consistent (i.e. having a model)! > > [quoted text clipped - 21 lines] > Whether it seems silly, it still is indeed the case, from the > definitions. Right. Thanks for pointing out these consequences of the definitions. > > > My terminology is 'true in a model' and it is precisely correct that > > > '2=1+1' is true in some models and not true in other models. Nam's [quoted text clipped - 13 lines] > > I affirm both. But not, I hope, without the parenthetical qualifications. That would be very confusing. > But the particular remark I made didn't mention PA. True. That is perhaps an unwarranted assumption on my part as to what was being talked about. > > It seems to be a simple ambiguity to be resolved. I personally would > > prefer (a) and would assume that you are using "model" in the sense [quoted text clipped - 12 lines] > > '2 = 1 + 1' is true in some models and false in other models No doubt it is false in some models of some theories, hence it is false in some models, QED. [...] > > Actually, I find it even more annoying when someone is being a jerk > > while being right! > > I guess that's a joke or something. Of course, it's subjective what > annoys a person. But it does seem disconnected to me to be more > annoyed by someone who is at least correct. I can assure you from long experience, I'd rather play chess with a poor loser than a swaggering, ungracious winner. -- hz SignaturePosted via a free Usenet account from http://www.teranews.com > For any structure S in a language L the structure assigns a truth-value > to every sentence of L. For every sentence phi of L, one of phi and [quoted text clipped - 7 lines] > The further question would be whether every structure is a model > of some non-null recursively axiomatized theory. Assuming again that not every sentence of L is a validity or its negation, take any contingent sentence phi as the sole axiom of theory T1, and take ~phi as the sole axiom of theory T2. Both theories are non-null and finitely axiomized, hence recursively axiomatized, and one of the two theories will be modeled by any structure for L. -- hz > Assuming again that not every sentence of L is a validity or its > negation, take any contingent sentence phi as the sole axiom of > theory T1, and take ~phi as the sole axiom of theory T2. Both > theories are non-null and finitely axiomized, hence recursively > axiomatized, and one of the two theories will be modeled by any > structure for L. Exactly. MoeBlee is SUCH an a.shole that HE thinks that THIS entitles him to say "therefore, every structure is a model, therefore I can say model when I mean structure, if I feel like it, and anybody who says I can't is proven wrong by this proof." I say instead that MoeBlee is proving that he is a flaming shitheel by choosing to use this argument. Of course, if I really felt confident that he was proving this about himself, then I wouldn't need to pollute my own posting record by belaboring the point with vulgarities. Let's just say that some points are sufficiently important that I feel the need to reinforce them. > > Assuming again that not every sentence of L is a validity or its > > negation, take any contingent sentence phi as the sole axiom of [quoted text clipped - 7 lines] > I can say model when I mean structure, if I feel like it, and anybody > who says I can't is proven wrong by this proof." Just for the record, that is not a quote of mine. Anyway, every structure is a model and every model is a structure. And in the literature, one may find contexts where one of the terms is used while in similar context elsewhere the other term is used. The unreasonable person is the one who makes TIRADES against the use of such innocuous phrasings. MoeBlee > The further question would be whether every structure is a model > of some non-null recursively axiomatized theory. The answer is yes, quite trivially: given a structure just take any finite, and hence recursive, set of sentences true in it. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus > > The further question would be whether every structure is a model > > of some non-null recursively axiomatized theory. > > The answer is yes, quite trivially: given a structure just take any finite, > and hence recursive, set of sentences true in it. Yes, thank you. At this point in the discussion, we have to add that one of the chosen sentences not be a validity. -- hz > For any structure S in a language L the structure assigns a truth-value > to every sentence of L. For every sentence phi of L, one of phi and [quoted text clipped - 4 lines] > > Perhaps you would like to clean that up a bit. It occurred to me Friday night that I spaced out the obvious and trivial: Let TH(M) be the set of sentences true in the model M. TH(M) is a complete, consistent theory and a proper superset of the set of validities. (I think what you wrote above might be along the same lines.) > The further question would be whether every structure is a model > of some non-null recursively axiomatized theory. Maybe just adopt one very simple axiom, such as an atomic sentence, or something like that? > > They are famous and important. Most famous in that way is perhaps the > > theory that is the set of sentences true in the standard model for the [quoted text clipped - 6 lines] > Yes, I've heard of that theory. It still seems a little imaginary to > me, not being recursively axiomatizable. It may seem imaginary to you, but it is an extremely important subject in mathematical logic. > > > Yuh, I guess the argument turns on whether you want to affirm > [quoted text clipped - 8 lines] > But not, I hope, without the parenthetical qualifications. > That would be very confusing. What is confusing? If one knows the definitions, both a) and b) are clearly true. model of a theory vs. model for a language. Those are clearly distinct. > > > Actually, I find it even more annoying when someone is being a jerk > > > while being right! [quoted text clipped - 5 lines] > I can assure you from long experience, I'd rather play chess with > a poor loser than a swaggering, ungracious winner. Ah, yes, chess does have its own special poignant psychology. MoeBlee > > For any structure S in a language L the structure assigns a truth-value > > to every sentence of L. For every sentence phi of L, one of phi and [quoted text clipped - 7 lines] > It occurred to me Friday night that I spaced out the obvious and > trivial: It happens. -- hz [...] > > > > Yuh, I guess the argument turns on whether you want to affirm > > [quoted text clipped - 11 lines] > What is confusing? If one knows the definitions, both a) and b) are > clearly true. Without the parenthetical qualifiers, a) and b) are clearly contradictory. > model of a theory > [quoted text clipped - 3 lines] > > Those are clearly distinct. Well, I guess context is important for an overloaded term. > > > > Actually, I find it even more annoying when someone is being a jerk > > > > while being right! [quoted text clipped - 7 lines] > > Ah, yes, chess does have its own special poignant psychology. Yes, winning is unanswerable. No need to slam the pieces down. -- hz > [...] > [quoted text clipped - 15 lines] > > Without the parenthetical qualifiers, a) and b) are clearly contradictory. Sorry, I mistakenly misread you. I thought you meant some other parenthetical qualifier needed beyond those already present. That was plainly my lapse. So, right, a) and b) are compatible as long as the parentheticals you included are included. MoeBlee >>>>> MoeBlee's assertion that it is a theorem that every structure is a >>>>> model of some theory is a new thought to me. [quoted text clipped - 120 lines] > that it is possible that Nam uses the word "model" in some sense > other than (a). For what it's worth, I actually don't use a different definition of the word "model" here, in FOL context. The phrase that seems difficult to some to understand is: (1) 'true relative to a model' Now suppose you believe PA is consistent and let M be a model of PA, then the following meta level statement would be true: (2) '2=1+1' is true in M. The question though, if we change the inference rules and come up with a different reasoning framework, say (FOL)', then would (2) still be true? The answer is "Not necessarily", depending on what changes of rules of inference that have taken place of course. Now if we consider the formal system PA as just a *collection of axioms* then syntactically PA is the same. But what we consider as M might or might not be an absolute model of PA, right? In that context then, '2=1+1' is being true in M is relative to the fact M might or might not be a model, depending on who's doing the reasoning. In that context, then the following statement would make sense: (3) '2=1+1' is true, relative to a model. which is not different in nature from the statement: (3') the speed of the train is 100 km/hour, relative to the framework M. At any rate, '2=1+1' is no more absolute than 'the speed of the train is 100 km/hour'. Those who believes otherwise would not realize the relativity nature of human mathematical reasoning (through FOL at least). Now there is another "more technical" way to demonstrate the model-relativity connoted in (3), and I did allude to this way a couple of times in the past. Basically, if we consider a FOL system sPA ("super" PA) whose languages contains *infinite* symbols: L(0, S,+,*,<, S',+',*',<', S'',+'',*'',<'', S''',+''',*''',<''', ...) In other words, besides symbol 0, the rest would be grouped together and each group, when coupled with 0, would form a language we could use to formalize a theory we'd name "PA". [This alone would signify the relativity of "PA" and associated models - and (3). Wouldn't it?] The (infinite number of) axioms of sPA would be the union of those individual axioms per each ("PA") group mentioned above. Now let's examine the anatomy of a model M of a general FOL formal system T. In a nutshell, the major components of M are: c1: a set S of individuals of which certain n-ary relations would exist. c2: a collection of n-ary relations, each of which would correspond to an n-ary symbol of the language. c3: collection of (subjective) interpretations, each of which would predicate a theorem-formula as true. Now MoeBlee suggested above: "Anyone may state definitions and explicate a discussion on the basis of those definitions." So in this context here and now, let's call S a "structure" (and temporarily forget if some text books reserve this word for something else). Then it's not hard to see that the relativity of a model M would come from component c3! For example, if the formula is "a < b", S = {a',b'}, and the n-ary relation is {(a',b')}, I still could at my subjective willingness interpret (or predicate) "a < b" as false while you or any other as true. And so relative to whose predicating or interpreting, M would be or *not* be a model of say T = {a < b}, or for that matter of T = {~(a < b)}. Now back to sPA, let S be the structure (i.e. the _set_) of individuals of a model of the integers (i.e. not the natural numbers). The long and the short of it is out of S, we know there exist _uncountably many_ "successor" functions S()'s, hence uncountably many n-aries "addition", "multiplication", and "less-than". Put if differently, not only S is a structure for sPA, it would be the very same structure for _uncountably many_ models of each "PA" theory (written in L(sPA)). But it's not hard to demonstrate that due to the subjective interpretation in c3, if one interpret S as a model of a "PA", others might disagree and (re)interpret S as *not* a model this "PA". Of course when we talk about "PA" we typically talk about it outside the context of this orangutan sPA system. But it should not matter! Given *any* L(PA), one could consider it as part of L(sPA). And given a model - over a structure S - of *any* "PA" theory one could interpret this structure S as a non-model of this "PA". In summary, a model is always (at least) *relative* to: - which exact theory in what exact language that's under consideration - which exact *subjective interpretation* (which would make a formula true). I know it's a bit long way to explain all this, and I don't think "typical" text books would care to give a discussion. But unless one could come up with some credible counter arguments, I'd think we'd have no choice but accept the relativity nature of reasoning in general. > For what it's worth, I actually don't use a different definition of the > word "model" here, in FOL context. The phrase that seems difficult to some [quoted text clipped - 6 lines] > > (2) '2=1+1' is true in M. We don't need 'belive' there. We could just say: If M is a model of PA, then '2=1+1' is true in M. > The question though, if we change the inference rules and come up with > a different reasoning framework, say (FOL)', then would (2) still be > true? Yes. The inference rules don't alter what sentences are true in what models. What the inference rules change is what theories we get. First order PA is DEFINED to be the set of sentences that are entailed in classical first order logic from the first order PA axioms. With a set of inference rules that doesn't yield the same theorems as classical first order logic you get a DIFFERENT theory from PA, you get HA (if the rules are intuitionistic logic) or whatever you want to name each DIFFERENT theory depending on a different logic. > The answer is "Not necessarily", depending on what changes of > rules of inference that have taken place of course. Now if we consider > the formal system PA as just a *collection of axioms* then syntactically > PA is the same. But we DON'T consider PA as just a collection of axioms. We consider PA to be the set of sentences entailed by that collection of axioms. (Or some people say it is the pair <A T> where A is the collection of axioms and T is the set of sentences entailed by A.) > But what we consider as M might or might not be an absolute > model of PA, right? What does "absolute model" mean? > In that context then, '2=1+1' is being true in M is > relative to the fact M might or might not be a model, depending on who's > doing the reasoning. In that context, then the following statement > would make sense: I'd grant that it depends on how we're doing the reasoning in the META- theory, since 'true in the model M' is defined in a meta-theory for first order PA and our proofs of whether something is true in the model are either done in the meta-theory or we rely upon the soundness theorem (which also is proven in the meta-theory) to infer that what is proven in the object theory is true in every model of the axioms of that object theory, as well as it is in the meta-theory that we prove that axioms are true in whatever models we claim the axioms to be true in. But that's not what you're talking about. Your present line of argument is, as usual, confused and ill-premised. > (3) '2=1+1' is true, relative to a model. > [quoted text clipped - 19 lines] > formalize a theory we'd name "PA". [This alone would signify the relativity > of "PA" and associated models - and (3). Wouldn't it?] NO! That's silly! Just CALLING something 'PA' doesn't show any relativity other than the quite prosaic sense we all already know that if you say "Sally Fields played the role of James Bond" then that is true if by 'Sally Fields' you are referring to the person Sean Connery. In some given overall mathematical context, such as a particular set theory to serve as a meta-theory, we DEFINE PA to be a certain exact mathematical object: a certain exact set of finite strings of symbols. Once we make that definition, it's just silly to worry about what happens if you say, "Oh, we get something different if we use 'PA' to stand for some other thing." > The (infinite number of) axioms of sPA would be the union of those individual > axioms per each ("PA") group mentioned above. [quoted text clipped - 3 lines] > > c1: a set S of individuals of which certain n-ary relations would exist. Okay, a non-empty set. > c2: a collection of n-ary relations, each of which would correspond to an > n-ary symbol of the language. Hmm, I suppose that's okay. But I'd rather say, a function that assigns to each n-ary predicate symbol of the language an n-ary relation on S. > c3: collection of (subjective) interpretations, each of which would predicate > a theorem-formula as true. YOU want c3 for whatever odd reason you do. I have no use for c3. Through c2 you had a nicely rigorous mathematical definition going, but then you obliterated it with the UNDEFINED terminology "subjective interpretations" and "would predicate a theorem-formula as true". > Now MoeBlee suggested above: > > "Anyone may state definitions and explicate a discussion on the basis > of those definitions." Sure. Though that was not intended to disallow that we may find certain definitions and explications to clash so strongly with ordinary terminology as to be irritating to work with. > So in this context here and now, let's call S a "structure" (and temporarily > forget if some text books reserve this word for something else). So, a 'structure' is defined by you now to be what we ordinarily call 'the universe' or 'the domain of discourse' of a structure. What is the advantage of you so confusingly switching these defintions? > Then it's > not hard to see that the relativity of a model M would come from component c3! Since c3 is UNDEFINED NONSENSE, it's hard to see what comes from it! > For example, if the formula is "a < b", S = {a',b'}, and the n-ary relation > is {(a',b')}, I still could at my subjective willingness interpret (or predicate) > "a < b" as false while you or any other as true. So who the L cares about that? We already have a formal mathematical definition of 'true in the model', but you'd rather we use some junky UNDEFINED nonsense about "subjective interpretation" instead. What possible advantage comes from that? > And so relative to whose > predicating or interpreting, M would be or *not* be a model of say T = {a < b}, [quoted text clipped - 9 lines] > if one interpret S as a model of a "PA", others might disagree and (re)interpret S > as *not* a model this "PA". Forget about whatever involves c3. If you'd have me adopt your c3, then I might as well have you adopt: A sentence is true in a model M iff one of the snails in my garden leaves, on the walkway, a trail more than 4 inches on the second Wednesday of the next month. > Of course when we talk about "PA" we typically talk about it outside the context > of this orangutan sPA system. But it should not matter! Given *any* L(PA), No, there is no "any" L(PA). L is an OPERATION. For each theory T, there is exactly ONE object that is L(T). For each theory, there is exactly one object that is THE language of that theory. > one could > consider it as part of L(sPA). No problem with L(sPA) as different from L(PA) if sPA and PA are different and happen also to have different languages. > And given a model - over a structure S - of *any* > "PA" theory one could interpret this structure S as a non-model of this "PA". > > In summary, a model is always (at least) *relative* to: In summary, apply c3, then check the snail trails each first Wednesday of the month, then take the Boolen product, then toss a coin for an arbitrary number of trials, then disregard having done all that, then declare the "relativity of predicating", then take deep inhalations from a bottle of model airplane glue and forget about everything whatsoever. MoeBlee > That's what's "a new thought to me": that an arbitrary structure > is a model of some non-null theory. I don't happen to know whether > that's true. Of course it's true. Every structure decides every sentence. Pick 1 sentence. Assert the theory with that 1 sentence decided the way the structure decides it as an axiom. Obviously, the structure is a model of that theory. But this depends rather sillily on how you go about describing a structure BEFORE you know what names the axioms are going to use, before you know what signature the language is going to have. Perhaps you have to attach that, first, too. > > > Shall we agree as to what sets of sentences constitute "a theory" first? > [quoted text clipped - 3 lines] > > closed under entailment (which, thanks to the completeness theorem, is > > a set of sentences closed under provability). That is the canonical definition. "Canon" meaning what it means, that is the ONLY definition you are going to get to use, without first explicitly attacking that definition. > > I adopt Enderton's definition. > > OK. It's NOT ok. It does violence to the word (theory). If you can't tell what's an axiom and you therefore can't tell even APPROXIMATELY *whether* some sentence is "in" (declared "provable") by the theory OR NOT, then you don't ACTUALLY HAVE any coherent *theory* (NON-local-technical sense) of what makes sentences "in" the theory true! > > > I usually think of "a theory" as having a recursive (or at least r.e.) > > > set of axioms. THAT *IS* the *CORRECT* definition. As MoeBlee is explaining, it is NOT the standard one. The point is that the standard is just broken. > > That's what's "a new thought to me": that an arbitrary structure > > is a model of some non-null theory. I don't happen to know whether [quoted text clipped - 3 lines] > Every structure decides every sentence. > Pick 1 sentence. Drat. You beat me to "publication". > Assert the theory with that 1 sentence decided the > way the structure decides it as an axiom. The structure could decide it's a false sentence. Picky, picky. > Obviously, the structure is a model of that theory. Or of the theory of the negation of that sentence. > But this depends rather sillily on how you go about > describing a structure BEFORE you know what names > the axioms are going to use, before you know what > signature the language is going to have. > Perhaps you have to attach that, first, too. Yuh, better pick a language/signature first, then define a structure for that. > > > > Shall we agree as to what sets of sentences constitute "a theory" first? > > [quoted text clipped - 27 lines] > As MoeBlee is explaining, it is NOT the standard one. > The point is that the standard is just broken. Yes, well, it's of a piece with tolerating arbitrary subsets of N or arbitrary functions from N to N or other "ideal" objects. Which I don't have a doctrinaire stand on. -- hz > > > > > I usually think of "a theory" as having a recursive (or at least r.e.) > > > > > set of axioms. [quoted text clipped - 6 lines] > arbitrary functions from N to N or other "ideal" objects. Which I > don't have a doctrinaire stand on. Although there is, perhaps a slight difference in "level": The ideal objects of various theories presuppose an underlying logic. Their existence follow from given axioms in a given logic. In this sense they are non-problematical. The assumption of the existence of non-recursively specified theories seems not to be the consequence of formally given axioms, but in consequence of meta-theoretical reasoning about the formalisms in which our logical theories are couched. -- hz > > > > > > I usually think of "a theory" > > > > > > as having a recursive (or at least r.e.) > > > > > > set of axioms. > > Yes, well, it's of a piece with tolerating arbitrary subsets of N or > > arbitrary functions from N to N or other "ideal" objects. With all due respect, you're missing the point. The point is TERMinological, NOT ONTOlogical. > > Which I > > don't have a doctrinaire stand on. Well, I do. You can prove, once you have enough machinery to prove that recursive and rec.enumerable subsets exist, that non-r.e. subsets must exist. Any definable class SMALLER than arbitrary, you can diagonalize out of. So that FORCES you (no pun intended) to "tolerate arbitrary". That is NOT the issue. > The assumption of the existence of non-recursively > specified theories IT is NOT an ASSUMPTION! It is a DEFINITION! > seems not to be the consequence of formally given axioms, Well, it certainly is, in the sense that you can prove, from relevant axioms, the existence of non-r.e. sets. > but in consequence of meta-theoretical reasoning about the > formalisms in > which our logical theories are couched. Just because it's meta-theoretical does NOT mean it is NOT ALSO *theoretical*, SIMULTANEOUSLY. The point being that the SAME ZFC that we study as an object theory is powerful enough TO SERVE as a meta-theory. So the meta- distinction here is moot. THE POINT is that "A theory" is WHATEVER WE SAY a theory is. The point is that we need to start TALKING differently. Currently, you get to have a theory by closing ANY old set of sentences under consequence, EVEN if you CANNOT SPECIFY EITHER of the "before" or "after" sets. "Theory" is arbitrary, so you need another term for theories actually worthy of the name. The generally encountered term for r.e. theories is "formal theory". We just need to redefine "theory" to denote what is currently aimed at by "formal theory" and come up with a more complicated term for "arbitrary" collections of sentences. > -- > hz > > > Authors such as Enderton take a theory to be any set of sentences > > > closed under entailment (which, thanks to the completeness theorem, is [quoted text clipped - 4 lines] > you are going to get to use, without first explicitly attacking > that definition. It's the one I use, but I don't know that it is canonical. Many authors use other definitions. > > > I adopt Enderton's definition. > > > OK. > > It's NOT ok. I thought you just aid it is the canonical definition. So it's not okay for me to use what you yourself regard as the canonical definition? And even if it is not canonical, it is at least quite common. Moreover, adopting a definition that is so common and also available to refer to in such a widely referenced book as Enderton's seems quite okay. > It does violence to the word (theory). > If you can't tell what's an axiom and you therefore can't > tell even APPROXIMATELY *whether* some sentence is "in" > (declared "provable") by the theory OR NOT, then you don't > ACTUALLY HAVE any coherent *theory* (NON-local-technical sense) > of what makes sentences "in" the theory true! To express what you just mentioned, we also have <A T> where A is an axiomatization of T. Granted, for many authors, a theory is an ordered pair <A T> where T is the set of consequences of A. If you prefer that definition or some other arrangement, then fine for you. But Enderton's definition is also handy for certain reasons (as evidenced by the way he uses it in his book). Live and let live should be the principle when it comes to such differences in definitions. Sure, one can argue that certain definitions are better than others, but your continual TIRADES about such matters are foolish. > > > > I usually think of "a theory" as having a recursive (or at least r.e.) > > > > set of axioms. > > THAT *IS* the *CORRECT* definition. > As MoeBlee is explaining, it is NOT the standard one. > The point is that the standard is just broken. Your continual DECREES as to what is "correct" are tiresome. Fine that you may give reasons for preferring definition D to definition D' (which should allow for understanding also the advantages of definition D' over definition D). But your continual HARANGUES and FIATS on such things as definitions are foolish. And please make up your mind: Most of the time you're telling people that they are wrong not to follow standard conventions (or the "paradigm", a word you use), and here you're saying it is NOT okay to do so. MoeBlee MoeBlee wrote: > To express what you just mentioned, we also have <A T> where A is an > axiomatization of T. Does "axiomatization of T" mean "recursive axiomatization of T"? Or can A be any old set of sentences? > Granted, for many authors, a theory is an ordered > pair <A T> where T is the set of consequences of A. -- hz > > To express what you just mentioned, we also have <A T> where A is an > > axiomatization of T. > > Does "axiomatization of T" mean "recursive axiomatization of T"? > Or can A be any old set of sentences? I think authors differ. As best I recall having read various authors, by 'axiomatization', some mean any set of sentences that entails T, while others mean a recursive set of sentences that entails T. Personally, I go with the first sense and use 'recursive axiomatization' for the second sense. MoeBlee > > Does "axiomatization of T" mean "recursive axiomatization of T"? > > Or can A be any old set of sentences? > > I think authors differ. It doesn't matter. Why you are so obsessed with what "authors" do is utterly beyond me. You can do anything you want as long as it is consistent. If the authors weren't smart enough to do likewise then that is their problem, not yours. > As best I recall having read various authors, > by 'axiomatization', some mean any set of sentences that entails T, > while others mean a recursive set of sentences that entails T. "As best I can recall" is simply neither adequate nor relevant. The issue is any case is not whether the AXIOMS are vs. aren't "any" set of sentences, BUT RATHER, whether THE THEORY is vs. isn't "any" set of sentences. There are 2 kinds of theories in the world, namely those that are r.e. and those that aren't. THAT is what matters. Just because that wasn't the question the questioner asked does NOT matter. The questioner doesn't always KNOW what matters. You should, though. > Personally, I go with the first sense and use 'recursive > axiomatization' for the second sense. You need some help from the dictionary. The "-ization" suffixes connote a process, a method, for actually finding or providing axioms in the theory. If the theory is not r.e. then this is basically not possible except in one of your usual trivial degenerate senses (like the one in which every structure is a model). The correct answer to the question is that if the theory is r.e. then it will have a recursive axiomatization, and if the theory is not r.e. then you have no hope of ever telling what a theorem is, let alone what an axiom is, so the alleged theory is hardly even worthy of the name. In other words, all axiom-sets that can reasonably or productively be thought of as axiom-sets are necessarily recursive. If the axiom-set is more complicated than r.e. to begin with, then obviously the usual rules of standard classical FOL are already in over their head. > > > Does "axiomatization of T" mean "recursive axiomatization of T"? > > > Or can A be any old set of sentences? [quoted text clipped - 4 lines] > Why you are so obsessed with what "authors" do is utterly > beyond me. I'm not obsessed with authors. What a silly remark by you. It's just that this subject is mainly disseminated through lectures and writings, and primarily, for those who are not gathered in a single lecture series, widely used textbooks provide the most common basis for definitions. I was asked what a term "means". I take that to be a request for what the term means to the people who use it as a technical term in the subject. So my best answer is to say what I know about how various authors use the term. > You can do anything you want as long as it > is consistent. If the authors weren't smart enough to do likewise > then that is their problem, not yours. Yes, I've said so myself that one can set up one's own system, and with one's own system of definitions, either keeping to the ordinary senses or departing from them. Usually, if one departs from ordinary senses, then I think one should have some reason for doing that. Indeed, authors differ among themselves, as they find convenient for the purposes of their own treatment. And I have culled from different books, and put together, in my typed notes, my own treatment (with much unfinished) with my own system of definitions, keeping to ordinary senses generally but tweaking sometimes to suit my own treatment. Moreover, do you at least see how what you're saying now about "you can do anything you want as long as consistent" goes against the grain of your continual harping ('harping' is putting nicely) that people are NOT within intellectual prerogative to go against the "paradigm" (as you've called it) and the presumed definitions? If I take you up now on your "do anything you want as long as consistent" and define, in some consistent system of definitions, a term so that it makes good sense but goes against the "paradigm", then I can pretty much count on you berating me for arrogating to myself a prerogative to go against the presumed definitions. Anyway, as I said, the poster asked what the term "means", so I take the context not to be what I stipulate the term means but what it means as ordinarily sed in the field of study. (And I did go on to say which meaning I adopt in my own terminology.) > > As best I recall having read various authors, > > by 'axiomatization', some mean any set of sentences that entails T, > > while others mean a recursive set of sentences that entails T. > > "As best I can recall" is simply neither adequate nor relevant. Oh, for godsakes. It's just a minor and casual disclaimer. I have never meant anyone as thoroughly and gratiutously disputatious as you. Do you have a disorder or something that causes you to need or thrive on that? > The issue is any case is not whether the AXIOMS are vs. aren't > "any" set of sentences, BUT RATHER, whether THE THEORY > is vs. isn't "any" set of sentences. You say, "THE issue" [emphasis added]. You do that, and similar, often. Maybe in this particular case you don't mean to be saying that the issue you wish to address makes the issue other people are addressing a distraction or aside from what is important, or similar, but it at least seems that way, and in certain other instances it has been that way. People like to discuss various aspects of these subjects. We don't need you to harp as to what "THE issue is". The poster asked me about a certain term, and I answered him. To do that, I don't need to check as to what George considers "THE issue" in a given discussion. > There are 2 kinds of theories > in the world, namely those that are r.e. and those that aren't. > THAT is what matters. And in certain other contexts people may wish to consider whether a certain set of sentences, - whether recursive or not - is a set that proves some other set of sentences. Please, give it a break already! > Just because that wasn't the question the questioner asked > does NOT matter. The questioner doesn't always KNOW what > matters. You should, though. PLEASE! I don't post to fully screen each question and remark as to what is THE matter of greatest import! That would be an absurd obligation to demand of anyone. There are lots of different items of discussion that get into a subject, ranging from trivial to small to important to, by SOMEONE's (YOURS?) ordering of importance. I, as do other reasonable people, feel well within bounds of reasonablity by fielding various items within that range as suits both my intellectual and recreational purposes for posting and without submitting to whatever hierarchy of importance YOU have decided upon at each and every context in an ever changing stream of contexts. > > Personally, I go with the first sense and use 'recursive > > axiomatization' for the second sense. > > You need some help from the dictionary. The "-ization" suffixes > connote a process, a method, for actually finding or providing > axioms in the theory. Again, how utterly captious of you. Mathematical terms don't always conform perfectly to dictionary or English grammar constraints. The word 'axiomatization' in mathematics is well established, and is clear in the sense I use it, and I can formalize it to any degree of formality required. > If the theory is not r.e. then this is > basically > not possible except in one of your usual trivial degenerate senses > (like the one in which every structure is a model). A theory T (as a set of sentences in a language such that the set is closed under entailment) is recursively axiomatized iff there exists a recursive set of sentences S in the language such that T is the set of consequences of S. That is a perfectly acceptable (and common) definition. > The correct answer to the question is that if the theory is r.e. > then it will have a recursive axiomatization, I correctly answered the question asked. And, as you said, if a theory is recursively enumerable then it has a recursive axiomatization. I have no issue with that. > and if the theory is > not r.e. then you have no hope of ever telling what a theorem is, > let alone what an axiom is, so the alleged theory is hardly even > worthy of the name. If the theory is not recursively enumerable then membership in the theory is not decidable. But that doesn't preclude discovery that certain sentences are or are not members of the theory. As to being "worthy" of being called a theory, we do understand that a theory that is not recursively enumerable does not fit certain informal notions of what a theory is or even other technical definitions of 'theory'. We don't promise that every technical meaning of a word such as 'theory' is faithful to every informal sense also or to other contrasting technical definitions of 'theory'. > In other words, all axiom-sets that can reasonably or productively > be thought of as axiom-sets are necessarily recursive. There is no problem in allowing a distinction between a recursive axiomatization and an axiomatization that is not recursive. > If the axiom-set is more complicated than r.e. to begin with, then > obviously the usual rules of standard classical FOL are already > in over their head. I'm not sure what you mean, but at least I do agree that for certain purposes, recursive axiomatizations are the ones we're interested in, as I have harped on that point in many a discussion with people who don't appreciate the importance of recursive axiomatization, but that doesn't preclude that it is reasonable to define 'theory' so that there are also theories that are not recursively axiomatizable, and that if you want to define 'theory' so that theory has a recursive axiomatization, then, fine, live and let live. For an extended technical discussion, we would have to agree on one definition or the other, but in a context of posting, I've given a clear and ordinary definition that serve the purpose especially as a conversation can be facilitated by agreement upon a definition that is the one used in what is arguably the most widely used and referenced textbook in the subject (or at least among the most widely used and referenced), and not with the argument that it is a good definition simply for being the one used in that textbook but rather that at least that textbook provides a common reference even as we may agree to depart from it or qualify it on certain definitions as suits our purpose. MoeBlee > Moreover, do you at least see how what you're saying now about "you > can do anything you want as long as consistent" goes against the grain > of your continual harping ('harping' is putting nicely) that people > are NOT within intellectual prerogative to go against the > "paradigm" (as you've called it) and the presumed definitions? You have this exactly backwards. *I* am going against the presumed definition in this case. > > Moreover, do you at least see how what you're saying now about "you > > can do anything you want as long as consistent" goes against the grain [quoted text clipped - 4 lines] > You have this exactly backwards. > *I* am going against the presumed definition in this case. No, you have it backwards that I have it backwards. Read what I wrote again. I AM recognizing that you are challenging a presumed definition. MoeBlee > > > Moreover, do you at least see how what you're saying now about "you > > > can do anything you want as long as consistent" goes against the grain [quoted text clipped - 8 lines] > again. I AM recognizing that you are challenging a presumed > definition. Just because I can have it both ways doesn't mean that you can. You have to approve of ONE or the other of these two sides that you idiotically allege me to be inconsistently taking. PLEASE PICK one. In any case, your whining about my "continual harping" in order to win a pissing contest here is not going to score any points. You have no idea who the judges/audience are. I have been here for over 20 years. Challenging at least a few pet peeved definitions is something I have been doing for most of that time. Obviously I therefore cannot believe that people cannot challenge these definitions. If you say I am proving, continually, that I believe that, then you simply don't know what is going on. > > > > Moreover, do you at least see how what you're saying now about "you > > > > can do anything you want as long as consistent" goes against the grain [quoted text clipped - 13 lines] > you idiotically allege me to be inconsistently taking. PLEASE PICK > one. No, you STILL have it backwards. Are you doing this on purpose? Usually you say (I'm paraphrasing), "You (whomever you're posting to) don't get to just choose your definitions; you have to follow the conventions that are in the literature and are presumed to be the standard in sci.logic." Then I told someone a couple of definitions that rival as the presumed ones and I stated which one I choose between those. Then you faulted me for relying on what is presumed (which contradicts your usual fulminations that we are to use the presumed definitions are not to presume that we can just make up our own). So you make it impossible for anyone to fulfill your requirements: If they take your latest instruction to feel free to make up their own definitions, then you excoriate them for doing that and not complying with the presumed defintions and paradigm; but then when I did comply with the presumed defintion, you criticized me for doing that! Moreover, I'm not obligated to pick between following always presumed defintion or always taking complete liberty to make my own ersatz definitions. I already explained that I usually follow presumed definitions unless there is some special reason (perhaps having to do with the specifics of the particular treatment I'm using) and that I choose among different definitions that are in the literature on the basis of various considerations (and also sometimes just to make a choice so that I can move on to substantive matters and not spend my life worrying whether definition D by author A is "better" than Defintion D' by author A', especially since usually D has its advantages and disadvantages relative to D' and vice versa). And I haven't said that you must not allow yourself the same prerogative. And you contradict yourself AGAIN, when you say that YOU may be reasonable to challenge presumed defintions but that I wouldn't be, since a few posts back you just criticized ME for NOT challenging a presumed defintion. Whether intentionally or not, you make reasonable conversation with you IMPOSSIBLE. Your posting verges on the insane. > In any case, your whining about my "continual harping" in order > to win a pissing contest here is not going to score any points. It's not whining. I'm telling you flat out, in your face, you're being a boob and a bore. And I've told you I'm not in any "pissing contest" with you. And I don't say things to you to "score points". > You have no idea who the judges/audience are. I have some idea. I have a pretty good appreciation of the range of people who post here and can guess as to the range of people who are reading. > I have been here > for over 20 years. How sad it's been twenty years of you posting in the really very very messed up way you do. > Challenging at least a few pet peeved definitions > is something I have been doing for most of that time. And I have no objection to you or anyone saying what they don't like about certain definitions, or proposing others, or reasonable things along those lines. > Obviously > I therefore cannot believe that people cannot challenge these > definitions. If you say I am proving, continually, that I believe > that, then you simply don't know what's going on. You don't recognize how very bizarre your postings are. MoeBlee > I'm not obsessed with authors. Of course you are. > It's just > that this subject is mainly disseminated through lectures and > writings, and primarily, for those who are not gathered in a single > lecture series, widely used textbooks provide the most common basis > for definitions. That is just out of touch with reality. If somebody is asking YOU what a term means then YOU are the ONLY basis for a definition. If you choose to defer to "widely used textbooks" then that is YOUR choice. > I was asked what a term "means". I take that to be No, really, you don't. > a request for what > the term means to the people who use it as a technical term in the > subject. So my best answer is to say what I know about how various > authors use the term. Unless you are actually going to evaluate/rank the competing uses, you are not contributing. > > I'm not obsessed with authors. > > Of course you are. Fatuous. > > It's just > > that this subject is mainly disseminated through lectures and [quoted text clipped - 7 lines] > YOU are the ONLY basis for a definition. If you choose > to defer to "widely used textbooks" then that is YOUR choice. Oh, come on! The poster asked me what a term means. I took that to be a request for a common definition of the term as it is used within the field of study, not just any old definition I might arbitrarily choose to make up. No matter what he had in mind by asking, without him qualifiying to say "What is your PERSONAL definition", it is utterly reasonalbe for me to take a question as to what a term means to be in the sense of how people in the field of study generally use the term. You're being an utter JERK for making any kind of deal out of the fact that I answered somebody that way. And not only did I tell him a couple of different definitions as generally used in the field of study, but I did also go on to add which of those definitions I personally adopt. You are, as usual, making yourself to be a complete a.s. > > I was asked what a term "means". I take that to be > > No, really, you don't. I didn't even finish the sentence and you cut me off before I even SAID what I "take", and you cut me off to say what I THINK (contrary to what I DO think). Are you TROLLING? Just arbitrarily disagreeing for the sake of it? > > a request for what > > the term means to the people who use it as a technical term in the [quoted text clipped - 3 lines] > Unless you are actually going to evaluate/rank the competing > uses, you are not contributing. Oh for GODSAKES! I don't in every post go on to cover every ramification of every matter raised. I answered the question. If asked for more of my opinion or more about the ramification of the differences in the definitions, then, my time, interest, and knowledge permitting I might have posted more in followup or even just say that I happen not to wish to delve into that particular matter (since my time, energy, knowledge, and enthusiasm is not unlimited to give exhaustive posts about every possible nook and cranny of this subject, especially when it is as to STIPULATIVE definitions). And that limitation doesn't disqualify the contribution of at least answering the question. Do you even have any IDEA of what an a.s your making of yourself? MoeBlee > > > I'm not obsessed with authors. > > > Of course you are. > > Fatuous. It's exemplified by your behavior. It's factually confirmed. Whether it is or isn't also "fatuous" is entirely moot, if it's factual. > > > It's just > > > that this subject is mainly disseminated through lectures and > > > writings, and primarily, for those who are not gathered in a single > > > lecture series, widely used textbooks provide > > > the most common basis > > > for definitions. Now THAT'S fatuous. BECAUSE it's factual. It's about as fatuous as responding to "what is your name?" with "The sky is blue." OF COURSE it is true that the textbooks say what they say. OF COURSE it is also true that what they say is itself also true. But EVERYBODY KNOWS THAT ALREADY. Wherefore it is fatuous (to put it charitably) to BELABOR any of it. > > That is just out of touch with reality. > [quoted text clipped - 3 lines] > > Oh, come on! No, dumbass, I will NOT come on: YOU come on. > The poster asked me what a term means. SO? There is such a thing as a stupid question. There is also such a thing as a question that overlooks something important or presumes something that, despite its conventionality, still NEEDS to be rejected. IF you are going to comment then YOU need to come on AND PAY ATTENTION to those sorts of things. > I took that to be a request It never matters what anybody is actually requesting. What matters is what they NEED. > > > > I'm not obsessed with authors. > [quoted text clipped - 5 lines] > It's factually confirmed. Whether it is or isn't also > "fatuous" is entirely moot, if it's factual. As I said, fatuous. That I have a strong interest in what various writers in the subject of mathematics have to say doesn't entail that I'm obsessed with authors. > > > > It's just > > > > that this subject is mainly disseminated through lectures and [quoted text clipped - 11 lines] > KNOWS THAT ALREADY. Wherefore it is fatuous (to put > it charitably) to BELABOR any of it. Because, YOUR ludicrous charge that I am obsessed is rebutted by mentioning the obvious. Since the primary way I learn about this subject is by authors, I am interested to note differences among authors so that I don't get confused by those differences and also can understand someone whose touchpoint is a different textbook or author. Wait, that is obvious too! Yes, such things are so obvious that it is amazing that you don't already see that they obviate your charge even before you make it! > > > That is just out of touch with reality. > [quoted text clipped - 15 lines] > IF you are going to comment then YOU need to come on > AND PAY ATTENTION to those sorts of things. I had no basis to think it was anything other than a legitimately posed question. And I do pay a reasonable amount of attention to context, but I do not accept your dicate that I may not also comment to a plain question with a plain answer whatever the context. > > I took that to be a request > > It never matters what anybody is actually requesting. > What matters is what they NEED. You still don't get it. I do not obligate myself to determine what is the exact thing each person most needs in every conversation. I am not a paid tutor (and I don't pretend that anyone would take me for any kind of tutor, either by my posting style or by the obvious limitations of my knowledge). No one is obligated by YOU or by anyone else to deliver responses that are only maximally tutorial. And, for that matter, as far as tutoring is concerned, you are about the worst I have ever read here: both in style and content. MoeBlee > You're being an utter JERK for making any kind of deal out of the fact > that I answered somebody that way. No, really, I'm not, and *I* am not the one making a deal out of this. The community's decision to let chaotic things qualify as theories IS ALREADY INHERENTLY a big (bad) deal. > And not only did I tell him a couple of different definitions as > generally used in the field of study, Well, that is not helpful. > but I did also go on to add > which of those definitions I personally adopt. This IS NOT A MATTER of PERSONAL adoption, DUMBASS. This is a GENERALLY BROADLY CONVENTIONALLY adopted definition! It is, for all its popularity, WRONG, but that does NOT legitimize or even POSSIBILITATE ANYone's having "a personal definition"! > You are, as usual, making yourself to be a complete a.s. You are still attacking me for no good reason. That may not be being a complete a.s but it is certainly nothing *I* need to tolerate. > > > I was asked what a term "means". I take that to be > [quoted text clipped - 3 lines] > SAID what I "take", and you cut me off to say what I THINK (contrary > to what I DO think). Oh, bullshit. I got down what you think JUST FINE. You can lie about it if you want to, but it won't help. Everybody who is actually reading can see. > Are you TROLLING? Just arbitrarily disagreeing for the sake of it? Since you're the one who's attacking me, this is simply a stupid question. I am disagreeing with the whole of standard practice on some things. You think I have any qualms about YOU? > > > So my best answer is to say what I know about how various > > > authors use the term. That is not usually ANYbody's BEST answer unless that was what was asked for. Even then, nobody actually wants to know the usage distribution. What they WANT to know is how THEY OUGHT to think about the concepts. Wherefore, as I said, > > Unless you are actually going to evaluate/rank the competing > > uses, you are not contributing. > Do you even have any IDEA of what an a.s your making of yourself? What you just did is called projection. It is a kind of madness even posting back to the kind of madness that on > > You're being an utter JERK for making any kind of deal out of the fact > > that I answered somebody that way. [quoted text clipped - 7 lines] > > Well, that is not helpful. Whatever YOU think of as helpful, it is a good thing to know what are the two of (or among them) most widely used defintions of a certain term. > > but I did also go on to add > > which of those definitions I personally adopt. > > This IS NOT A MATTER of PERSONAL adoption, DUMBASS. > This is a GENERALLY BROADLY CONVENTIONALLY adopted > definition! You really are ZONKED aren't you? There were two different definitions, both in wide use. I told the poster that. Then I told the poster which one I use. That serves the purpose of letting the poster understand me exactly when I use the term. First you faulted me for just reciting the ordinary technical definitions and not taking the initiative to propose my own definition. Then I told you why I stated the ordinary definitions, and also that I did go on to state which one I use. Now you're back full circle faulting me for being so bold as to even choose between established definitions, which is not even as bold as stipulating a definition of my own devising or stipulating that I adopt some definition that defies the ordinary ones, which is what you faulted me for NOT doing! You might as well hold up two fingers and ask me to confirm that you're holding up two fingers then zap me with electricity for confirming, then ask me to confirm again then zap me for NOT confiming. So, are you at least slightly insane or are you just enjoying this as a psychological game of perpetually contradicting even if it means perpetually contradicting yourself. > It is, for all its popularity, WRONG, but that does NOT > legitimize or even POSSIBILITATE ANYone's having "a personal > definition"! You just flat out contradicted what you said to me: You faulted me for NOT taking the initiative to propose a definition that defies the ordinary technical ones! And "personal definition". I just said which of the ordinary defintions I use. My purpose in studying mathematics is not to brood over every defiinition as to how it conforms or not to some common informal sense of the word outside mathematics. > > You are, as usual, making yourself to be a complete a.s. > > You are still attacking me for no good reason. > That may not be being a complete a.s but it is > certainly nothing *I* need to tolerate. I could hardly have made my reasons any more forcefully clear. > > > > I was asked what a term "means". I take that to be > [quoted text clipped - 7 lines] > You can lie about it if you want to, but it won't help. > Everybody who is actually reading can see. Beautiful. Not only are wrong about what I said. You are wrong about what I think. And you are wrong that I am lying as to what I think. And you are also claiming that everybody else agrees with you about that. > > Are you TROLLING? Just arbitrarily disagreeing for the sake of it? > > Since you're the one who's attacking me, this is simply a stupid > question. I am disagreeing with the whole of standard practice > on some things. You think I have any qualms about YOU? Hilarious. As if your remarks have not been directed to me personally. Wow. > > > > So my best answer is to say what I know about how various > > > > authors use the term. [quoted text clipped - 3 lines] > wants to know the usage distribution. What they WANT > to know is how THEY OUGHT to think about the concepts. Now you are speaking for the poster. Maybe his question was a request to be told what he ought to think, but I have no reason to think that it was. I took his question at face value. He asked me what is meant by 'theory' (or whatever his exact wording of the question), and I told him a couple of the more prominent defintions, and told him which one I use (so that he would know what I mean when I use the word). And for my doing that, you tore off on a bizarre attack. > Wherefore, as I said, > [quoted text clipped - 3 lines] > > What you just did is called projection. How ironic that you said that. MoeBlee > > > Does "axiomatization of T" mean "recursive axiomatization of T"? > > > Or can A be any old set of sentences? > > > > I think authors differ. [...] > > As best I recall having read various authors, > > by 'axiomatization', some mean any set of sentences that entails T, > > while others mean a recursive set of sentences that entails T. [...] > Just because that wasn't the question the questioner asked > does NOT matter. The questioner doesn't always KNOW what > matters. The point of my question, such as it was, is that the first sense of the definition reduces to Enderton's definition. -- hz > The point of my question, such as it was, is that the first sense > of the definition reduces to Enderton's definition. You cut too much. Definition OF WHAT term? If the term was "axiomatization" then THE point, REGARDLESS of YOUR point, is that you don't GET to have questions about THAT definition until AFTER resolving the PRIOR question about the definition of the term "Theory". This is ESPECIALLY true when the question is YOUR question, the point being that the "arbitrary xor r.e.?" question IS ALREADY relevant for theories. Anytime anybody asks that question about anything else, you almost have to refer them back to the theory question FIRST. The resolution there IS likely to affect the resolution in your other realm. Of course "axioms" being a SUBrealm of "theory" in any case, the likelihood is maximized. > MoeBlee's point was that people were in a very generalized habit of > using > "model" withOUT requiring it to be a model of anything in PARTICULAR. No that was NOT my point. I didn't say anything about a generalized habit or any habit. And I did not say that M can be a model without being a model of some set of sentences. As to the word 'particular', I simply observed that we have the 2-place predicate: M is a model of G and that we can also have the 1-place predicate M is a model by defining M is a model iff there is a set of formulas G such that M is a model of G. And doing so harms nothing. And doing so helps to see why such as Chang & Keisler say such things as M is a model for the language L where other authors say M is a structure for the language L. Chang & Keisler's use is clear since 'model' and 'structure' can be be used interchangably in such contexts since M is a model iff M is a structure. > That is an observable fact about the brute weight of usage; neither I > nor [quoted text clipped - 13 lines] > intellectually > acceptable behavior. There is nothing misleading or damaging to mathematics by my usage, especially as I defined it explictily and as it is used even in such an authoritative works as Chang & Keisler (I mean the kind of thing as 'M is model for the language L' ). MoeBlee > > MoeBlee's point was that people were in a very generalized habit of > > using "model" withOUT requiring it to be a model > > of anything in PARTICULAR. > No that was NOT my point. It WAS SO TOO, DUMBASS. And LIAR, too, for that matter. > I didn't say anything about a generalized > habit or any habit. sh.t. You cited Enderton by way of trying to prove your point. You cited it implying that EVERYbody was entitled to have this habit. You certainly now claim that Chang, Keisler et al have the habit. > And I did not say that M can be a model without > being a model of some set of sentences. Of course not -- you proved the precise opposite. You proved that every structure is a model of some sentence, so every structure is "a model". So therefore, you are not "wrong" to call something a model WITHOUT ASSOCIATING IT WITH ANY PARTICULAR SENTENCES. Wherefore, you may kindly go f.ck yourself. Seriously, You wasted a whole thread proving that Enderton (of all people!) entitled you to talk about models in general, and you actually proved that models COULD be referred to as models without referring to any particular set of sentences. And now you try to DENY it?? This is pathetic. > As to the word 'particular', SHUT *UP*! STOP splitting hairs! YOU KNOW what the argument was about and EVERYBODY SAW who took which side! You simply have no leg to stand on. > I simply observed that we have the 2-place predicate: > [quoted text clipped - 3 lines] > > M is a model No, really, we can't. Your CONTINUING to defend THIS BULLSHIT ENTIRELY REFUTES your claim above about "I never said M could be a model without being a model of some set of sentences". This entire locution is EXACTLY what I was talking about. That you chose to interpret it contrarily simply proves you are an a.shole. > by defining > > M is a model "Model" ALREADY HAS a definition, dumbass. > iff there is a set of formulas G such that M is a model > of G. > > And doing so harms nothing. It DOES SO TOO harm something, DUMBASS. I did EXPLAIN what it was. If you choose to believe that it harms nothing then I choose to believe you are WAY TOO STUPID to be worth bothering with. Actually, I don't believe you're stupid in any case. I believe you know perfectly well what is going on here with the predicates and that your actual motivation for publishing this sh.t, phrased as tight little logical proofs, is to publicly humiliate me. It can't work. The fact that you are proving that you don't know THAT is actually quite gratifying. > And doing so helps to see why No, it doesn't. Doing so is just stupid, period. > such as Chang & Keisler say such things > as > > M is a model for the language L That IS JUST STUPID. > where other authors say > > M is a structure for the language L. Other authors say that BECAUSE the previous version is SO stupid that NOT EVEN ITS BLESSING BY C&K can save it. THAT is what THEY understand but YOU don't. > Chang & Keisler's use is clear No, it isn't. > since 'model' and 'structure' can be be > used interchangably No, they can't; that's the whole point. > in such contexts Dipshit: THIS IS NOT a matter of CONTEXT! It is ALWAYS the case in ALL contexts that "model" NEEDS to be a model OF something! > since M is a model iff M is a structure. You continue to cite that is though it were important and defensible. There is no actual defense for that behavior, or for the locution either (root cause). > There is nothing misleading or > damaging to mathematics by my usage, Of course there is. > especially as I defined it explictily Dipshit: YOU DON'T GET to define *model*! Model ALREADY HAS a meaning! > and as it is used even in such > an authoritative works as Chang & Keisler As you have already noted, OTHER AUTHORS DISAGREE. THERE IS A REASON for that. But appeal to authority is a fallacy in EITHER direction (yours OR mine). You have to just look at it for yourself and decide whether YOU think the distinction is worth observing. And the answer to THAT is OF COURSE you would, IF you didn't have the insane need to win this public argument. Which, for the record, YOU CAN'T. > > > MoeBlee's point was that people were in a very generalized habit of > > > using "model" withOUT requiring it to be a model [quoted text clipped - 4 lines] > It WAS SO TOO, DUMBASS. > And LIAR, too, for that matter. Classic George. Rebuttal by mere declamation, and as to ANOTHER person's INTENTIONS. > > I didn't say anything about a generalized > > habit or any habit. > sh.t. > You cited Enderton by way of trying to prove your point. > You cited it implying that EVERYbody was entitled to > have this habit. You certainly now claim that Chang, > Keisler et al have the habit. I cited them to note certain similarities and differences as to certain definitions. I haven't said anything or suggested anything or inteneded anything about "habits". I'm not responsible for what YOU fabricate in your own mind as to what points I am trying to convey. > > And I did not say that M can be a model without > > being a model of some set of sentences. [quoted text clipped - 5 lines] > a model WITHOUT ASSOCIATING IT WITH ANY > PARTICULAR SENTENCES. Whatever you mean by "associating with any particular sentences", what I did is use existential generalization. If S is structure, then there exists a set of sentences T such that S is a model of T. Therefore every structure is a model of some set of sentences. And, if 'S is a model' is defined as 'There is a set of sentences T such that S is a model of T', then every structure is a model. > Wherefore, you may kindly go f.ck yourself. > Seriously, > You wasted a whole thread proving that Enderton > (of all people!) entitled you to talk about models > in general, No, YOU'VE wasted more than you deserve by making an attack on what was plain and clear. > and you actually proved that models > COULD be referred to as models without referring > to any particular set of sentences. > And now you try to DENY it?? You just get more and more bizarre. I haven't used the word 'particular' as you are doing. YOUR use of the word 'particular' doesn't confound what I said: I said, M is a model iff M is a structure and there is a set of sentences true in M. And YOUR response right here is to my remark: "And I did not say that M can be a model without being a model of SOME set of sentences." [emphasis added] So whatever knots your getting yourself tied up in regarding 'particular' are all your own knots. For a structure S to be a model, of course, S must be a model of SOME ('some' in the ordinary sense in mathematics of 'at least one') set of sentences. And, as it turns out, every structure IS a model of some set of sentences. > This is pathetic. > [quoted text clipped - 4 lines] > YOU KNOW what the argument was about and > EVERYBODY SAW who took which side! No, because YOU are using the word 'particular' in a way I haven't. What I said was exact and I specifically chose the word 'some' or 'exists' for that purpose. If you make a mess of something with the word 'particular', then that is YOUR problem, not mine. And that is not splitting hairs. > You simply have no leg to stand on. > [quoted text clipped - 12 lines] > your claim above about "I never said M could be a model > without being a model of some set of sentences". You're still completely mixed up. M is a model iff there is some set of sentences that M is model of. One more time, since you are being utterly obtuse, I am NOT contradicting myself: M is a model of T iff (M is a structure & T is a set of sentences & all members of T are true in M) [a 2--place predicate of M and T] M is a model iff there exists a T such that M is a model of T [a 1- place predicate of M] And so, if M is a model then there is some set of sentences T such that M is a model of T. > This entire locution is EXACTLY what I was talking about. > That you chose to interpret it contrarily simply proves [quoted text clipped - 17 lines] > then I choose to believe you are WAY TOO > STUPID to be worth bothering with. You've not shown what is harmed. > Actually, I don't believe you're stupid in any case. > I believe you know perfectly well what is going on > here with the predicates and that your actual motivation > for publishing this sh.t, phrased as tight little logical > proofs, is to publicly humiliate me. Wow. That is at least bordering on paranoia. I don't post formulas and proofs to humilate you. I posted them in this instance to initially explain a minor point of terminology and then to defend from your confused and bizarrely extended attacks on the matter. > It can't work. > The fact that you are proving that you don't know THAT [quoted text clipped - 4 lines] > No, it doesn't. > Doing so is just stupid, period. You're so full of denunciation that you denounce even a bit of a sentence that doesn't even say anything onto itself without the rest of the sentence. > > such as Chang & Keisler say such things > > as > > > M is a model for the language L > > That IS JUST STUPID. Nothing stupid about it. > > where other authors say > [quoted text clipped - 4 lines] > BY C&K can save it. THAT is what THEY understand > but YOU don't. That authors vary in terminology doesn't entail that they think of each other's different terminology as stupid. > > Chang & Keisler's use is clear > > No, it isn't. Sure it is. They define it. > > since 'model' and 'structure' can be be > > used interchangably [quoted text clipped - 6 lines] > It is ALWAYS the case in ALL contexts that "model" > NEEDS to be a model OF something! Every model is a model of some set of sentences. And it is perfectly sensible also to say "M is a model" without saying which sets of sentences M happens to be a model of. There might not be much advanatage or motivation just to say "M is a model", except in at least one instance, which is the one where I pointed out that M is a model iff M is a structure, which was just a sidenote to my original point (not about theories but about languages) that "M is a structure for the language L" and "M is model for the language L" are equivalent. > > since M is a model iff M is a structure. > > You continue to cite that is though it were important and > defensible. There is no actual defense for that behavior, > or for the locution either (root cause). I gave the defense. You only argued against it by getting yourself completely mixed up over the word 'particular'. > > There is nothing misleading or > > damaging to mathematics by my usage, > > Of course there is. You've not shown it. > > especially as I defined it explictily > > Dipshit: YOU DON'T GET to define *model*! > Model ALREADY HAS a meaning! I didn't redefine the meanings of "M is a model of for the language L" nor of "M is a model for the set of sentences T". What I did was to be explicit that when I say "M is model" I mean "M is a model of some set of sentenes". > > and as it is used even in such > > an authoritative works as Chang & Keisler > > As you have already noted, OTHER AUTHORS DISAGREE. > THERE IS A REASON for that. Authors on all kinds of subjects in mathematics have different usage. That can be for many reasons, including that some usges may be better than others in certain ways, or that certain usages better fit an author's particular treatment, or that the author is not so very fussy about certain usage, et. al. That authors have different usage does not entail that there is even controversy present. > But appeal to authority is > a fallacy in EITHER direction (yours OR mine). I am not appealing to authority to establish a fact about the world or even line of reasoning. I just cite the authority to establish that the usage is present prominently in the literature and that, to the extent one adopts usage on the basis of the authoritativeness of the author, then Chang & Keisler are authoritiative on the subjet of model theory. > You have to just look at it for yourself and decide whether > YOU think the distinction is worth observing. > And the answer to THAT is OF COURSE you would, > IF you didn't have the insane need to win this public > argument. Which, for the record, YOU CAN'T. I have a system of definitions that I have put together based primarily on certain textbooks. I decide upon each definition based on various considerations and I spend only as much time on that decision as I think is reasonable (there's a lot of mathematics to learn; I can't brood for long periods of time on every single definition, as long as at least each is in correct definitional form). Meanwhile, I don't begrudge others from devising their own system or stating which definitions they adopt from among differing definitions in the literature. And I don't begrude someone for giving reasons for why certain defintions, even as they are stipulative, have certain advantages and disadvantages, or even from proposing entirely new definitions and overthrowing certain old ones. But that I am DEFENDING the quite reasonable usage I've stated from your completely confused and boorish attacks does not make me stupid, willfully stubborn, unreasonable, looking for a "pissing match", or out to humilate you. MoeBlee > That IS JUST STUPID. Right. Chang & Keisler's definition is STUPID, the downward Löwenheim-Skolem theorem shows that the upward Löwenheim-Skolem theorem is stupid and so on. Perhaps you're using "stupid" in some technical sense I'm unaware of? SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus > > That IS JUST STUPID. On Jan 10, 5:20 am, Aatu Koskensilta <aatu.koskensi...@xortec.fi> wrote: > Right. Chang & Keisler's definition is STUPID, the downward Löwenheim-Skolem > theorem shows that the upward Löwenheim-Skolem theorem is stupid and so on. > Perhaps you're using "stupid" in some technical sense I'm unaware of? I wish I could reply that perhaps you're just being an a.shole, but the "perhaps" part would be inaccurate. You can engage the philosophical point or not. But if you choose not to, groundless invective is not relevant. > ... a.shole .... invective is not relevant. George taking Aatu to task for his verbal style: Ladies and gentlemen, I present your Jaw-Droppingly Hypocritical Usenet Moment for the day. Marshall > But if you choose not to, groundless invective is not relevant. It seldom is. Realising this, why not mend your silly ways? SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus > Wait a second. Aren't those axioms /true/ only when /interpreted/ > (i.e. in a model)? Hence isn't it the /interpretation/ that assigns > truth to an axiom? NO, DUMBASS. It is the designation of an axiom AS an axiom that assigns PROVABILITY to the axiom. Structures and interpretations don't have to be invoked AT ALL. THAT is the point. IF they are then truth and provability have to be DISTINGUISHED, but the whole point is that the whole structural/semantic piece IS SHAVABLE WITH OCCAM'S RAZOR. It does NOT have to be ackonowledged AT ALL. >> Wait a second. Aren't those axioms /true/ only when /interpreted/ >> (i.e. in a model)? Hence isn't it the /interpretation/ that assigns >> truth to an axiom? >> > It is the designation of an axiom AS an axiom that assigns > PROVABILITY to the axiom. That's right. But that was not my question. :-) (I asked for TRUTH, not PROVABILITY.) F. SignatureE-mail: info<at>simple-line<dot>de > Prof.Smith and I have been talking, for example, about the > intended model vs. the formal language. He is the one who > said that he didn't think formal languages should even be referred > to as a language. That is considerably less defensible than anything > *I* have ever said. What I said was that uninterpreted syntax is just that, uninterpreted. So not (yet) a vehicle for communicating anything. And so, in the ordinary sense of the term, not a language. Hardly an indefensible view. If some logicians, and according to George, many/most computer scientists do talk of uninterpreted syntax as a language (without qualification) then fine as long as the jargon is made clear: but it *is* in that usage specialist jargon, and not -- to my mind -- entirely happy jargon as it is potentially misleading in various ways. I hesitate to add "as evidenced here". >> Prof. Smith and I have been talking, for example, about the >> intended model vs. the formal language. He is the one who [quoted text clipped - 10 lines] > scientists do talk of uninterpreted syntax as a language (without > qualification) Which indeed is the case. See for example http://plato.stanford.edu/entries/logic-classical/ "Typically, a /logic/ consists of a formal or informal language together with a deductive system and/or a model-theoretic semantics. The language is, or corresponds to, a part of a natural language like English or Greek. The deductive system is to capture, codify, or simply record which inferences are correct for the given language, and the semantics is to capture, codify, or record the meanings, or truth-conditions, or possible truth conditions, for at least part of the language." > then fine as long as the jargon is made clear: but it > *is* in that usage specialist jargon, and not -- to my mind -- > entirely happy jargon as it is potentially misleading in various ways. > I hesitate to add "as evidenced here". Well, how about the following approach? Those /formal languages/ (considered in formal logic) are (usually) at least "capable" of "bearing meaning". F. SignatureE-mail: info<at>simple-line<dot>de > >> Prof. Smith and I have been talking, for example, about the > >> intended model vs. the formal language. He is the one who > >> said that he didn't think formal languages should even be referred > >> to as a language. That is considerably less defensible than anything > >> *I* have ever said. Prof Smith how dishonestly defends: > > What I said was that uninterpreted syntax is just that, uninterpreted. > > So not (yet) a vehicle for communicating anything. Yes, you did indeed say that, but that is not germane to the point I was just making, which is that you ALSO said, > you are thinking of a formalized language as a purely syntactic system. > Why? I know that logic books occasionally talk like this, but this has > always seemed to me simply to be a misuse of the word "language". THAT is the point under debate. I offered what should've been a clear defense, which is simply that on the computing side, people do legitimately care about languages as purely syntactic collections and do legtimately care about how hard they are to define/parse. It is a legitimate arena of inquiry. It is normal to talk and think about purely formal languages AND TO CALL THEM LANGUAGES. It is not "simply a misuse of" ANYthing. The fact that you choose to call it such makes you simply abusive. > > And so, in the ordinary sense of the term, > > not a language. YOU DIDN'T SAY "In the ordinary sense of the term". YOU DID SAY "simply a misuse". Simply choosing a more restricted technical meaning for a term in a context of discourse IS NOT simply misusing the term. > > Hardly an indefensible view But hardly recognizable as the view were promoting. > > If some logicians, and according to George, many/most computer > > scientists do talk of uninterpreted syntax as a language (without > > qualification) Hardly. We call them formal languages. But just as surely as a yellow banana is a banana, a formal language is a language. > Which indeed is the case. FF is being stupid here. He is claiming to agree with you while citing an article that agrees with YOUR point, but DISagrees with what people who were studying complexity-classes of formal languages would do. > See for examplehttp://plato.stanford.edu/entries/logic-classical/ > [quoted text clipped - 5 lines] > is to capture, codify, or record the meanings, or truth-conditions, or > possible truth conditions, for at least part of the language." This is all old by now and it does not matter what was typical in 1951. Nowadays we have clear definitions of all this. > > then fine as long as the jargon is made clear: but it > > *is* in that usage specialist jargon, and not -- to my mind -- > > entirely happy jargon as it is potentially misleading in various ways. > > I hesitate to add "as evidenced here". Well, good hesitation. My point is simply that Prof.Smith's contention that formal languages are too degenerate to be really called languages is fundamentally ignorant. He is not a linguist. His claim that they need to be interpreted in order to communicate is factually false and every decidable categorical first- order theory constitutes a living counter-example to it. The fact that is currently debating against that says a lot more about his character than it does about his intellect. > Well, how about the following approach? Those /formal languages/ > (considered in formal logic) are (usually) at least "capable" of > "bearing meaning". That is not under contention. If you interpret them then of course they have meaning AFTER that. My point was simply that you don't have to do anything SEMANTIC in order to CAUSE them to "realize" their "capability" of bearing meaning BECAUSE FIRST-ORDER CONSEQUENCE IS SYNTACTIC, because there is a completeness theorem. >> See for examplehttp://plato.stanford.edu/entries/logic-classical/ >> [quoted text clipped - 7 lines] >> > This is all old by now and it does not matter what was typical in 1951. Huh? Hey, george, this was written in 2000 (!), by Stewart Shapiro. ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ > Nowadays we have clear definitions of all this. You may be pretty sure that this guy knows what he's talking about. F. SignatureE-mail: info<at>simple-line<dot>de > >> See for examplehttp://plato.stanford.edu/entries/logic-classical/ > [quoted text clipped - 9 lines] > > Huh? Hey, george, this was written in 2000 (!), by Stewart Shapiro. G. Frege, I basically agree with you and Peter that Peter is well within his intellectual prerogative to give a stipulative definition of 'formal language' so that a formal language has both a formal syntax and formal semantics, which also is in general lines of agreement with the Church quote you gave. But, just to be clear, the above quote doesn't say that a language has both a syntax and semantics. Rather, the above quote says that a LOGIC has a language, a deduction system, and a semantics. MoeBlee >> Huh? Hey, george, this was written in 2000 (!), by Stewart Shapiro. >> [quoted text clipped - 6 lines] > semantics. Rather, the above quote says that a LOGIC has a language, a > deduction system, and a semantics. Sure. I now that. I posted it to back up /george's/ position (just to be fair). Here's the quote with some context: ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ > If some logicians, and according to George, many/most computer > scientists do talk of uninterpreted syntax as a [formal] language > (without qualification) ... Which indeed is the case. See for example http://plato.stanford.edu/entries/logic-classical/ "Typically, a /logic/ consists of a formal or informal language together with a deductive system and/or a model-theoretic semantics. The language is, or corresponds to, a part of a natural language like English or Greek. The deductive system is to capture, codify, or simply record which inferences are correct for the given language, and the semantics is to capture, codify, or record the meanings, or truth-conditions, or possible truth conditions, for at least part of the language." ~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~~ It's indeed a fact that "we" (today?) _do_ talk about "formal languages" without taking into account any semantical considerations. In brief: "[...] a formal language is a recursively defined set of strings on a fixed alphabet." http://plato.stanford.edu/entries/logic-classical/ george: "There simply now IS a restricted sense of "formal language" in which certain sets of strings are formal languages." --- You see, he's right. F. SignatureE-mail: info<at>simple-line<dot>de > > G. Frege, I basically agree with you and Peter that Peter is well > > within his intellectual prerogative to give a stipulative definition [quoted text clipped - 7 lines] > Sure. I now that. I posted it to back up /george's/ position (just to be > fair). Ah, that makes sense. > george: "There simply now IS a restricted sense of "formal language" in > which certain sets of strings are formal languages." > > --- You see, he's right. Of course we know that there are senses in which a formal language is a set of strings, also a sense in which the language is taken not to be the set of strings but rather a tuple that "encodes" a certain signature and sets of kinds of symbols (cf., e.g. Monk's textbook, which, for its elegance and rigor, I think is one of the best definitions of 'first order language'), and other senses of language as purely syntactical. But there are other notions (even if more in the minority these days) in which a language is a syntax and a semantics. And even of a language as a syntax, semantics, and pragmatics. And perhaps other conceptions of 'language' and 'formal language'. So, as long as Peter is clear that he's just giving his definition for the purpose of conveying his exposition, I see nothing wrong in that, and more power to him. MoeBlee > Of course we know that there are senses in which a formal language is > a set of strings, also a sense in which the language is taken not to [quoted text clipped - 12 lines] > the purpose of conveying his exposition, I see nothing wrong in that, > and more power to him. I guess this (your account) is a reasonable approach. george, on the other hand, originally claimed: "Prof. Smith and I have been talking, for example, about the intended model vs. the formal language. He is the one who said that he didn't think formal languages should even be referred to as a language. That is considerably less defensible than anything *I* have ever said." And _that_ (i.e. george's last claim) is certainly not reasonable (imho). F. SignatureE-mail: info<at>simple-line<dot>de > > Of course we know that there are senses in which a formal language is > > a set of strings, also a sense in which the language is taken not to [quoted text clipped - 25 lines] > And _that_ (i.e. george's last claim) is certainly not reasonable > (imho). Smith gave arguments (whether I agreee with them or not) for his view that 'langauge' is not suited to refer to systems that are merely syntactical. He also allowed that it's fair enough though that the word is used stipulatively (as long as the stipulation is clearly given) in that way, though he thinks it not best. Meanwhile, it is a fact that some authors in mathematical logic do use 'language' in the way Smith prefers and it is my view that that usage too is fair enough as long as its stipulation is clearly given too. MoeBlee > >> "Typically, a /logic/ consists of a formal or informal language together > >> with a deductive system and/or a model-theoretic semantics. The language [quoted text clipped - 4 lines] > >> possible truth conditions, for at least part of the language." > Huh? Hey, george, this was written in 2000 (!), by Stewart Shapiro. In which case I completely agree with it. He is talking about what is typical. This is how it is usually done, yes. This is expository. It is not relevant to the specific formal enterprise that was under debate. And it was cited BY OTHER PEOPLE as MATCHING something that WAS promulgated TECHNICALLY in 1951. 50 years later, everybody's perspective is different. In any case, none of THIS is the POINT! The POINT is that natural language IS NOT RELEVANT PERIOD to this whole enterprise! > > Prof.Smith and I have been talking, for example, about the > > intended model vs. the formal language. He is the one who [quoted text clipped - 13 lines] > entirely happy jargon as it is potentially misleading in various ways. > I hesitate to add "as evidenced here". I just want to mention that the FOL theories under discussion don't come as entirely uninterpreted syntax. The logical operators (sentence connectives, quantifiers) come pre-installed with meaning. The "interpretation", in one technical sense, is confined to equipping the predicates with reference, and supplying a domain of discourse for that purpose. If the language contains constants and/or function symbols, they too will have to be "interpreted" to have a fully interpreted language. If you consider the sign for equality a logical symbol, its meaning will be conferred by the axioms concerning it. -- hz > If you consider the sign for equality a logical symbol, its meaning > will be conferred by the axioms concerning it. If it is a *logical* symbol then it will necessarily be *prior* to any axioms. In FOL with equality, substitutivity in particular *should* be classified as a rule of inference AS OPPOSED to an axiom-schema. The Logic is characterized by its ROI's and in FOL *with* equality, the equality is part of the logic. > You are beginning by presuming the existence of an intended model. > > That is just not the way it's generally done any more. Mathematicians don't think, or do not begin by thinking, the natural numbers -- or the reals, or sets of sets of sets of reals, or what have you -- exist and they can state and proves stuff about them? A very curious idea. > The completeness theorem is fundamentally a proof that first-order > semantics SIMPLY DOESN'T EXIST. Right. Just as the fundamental theorem of analysis establishes no-one knows how to ride a bicycle. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus On Dec 19, 7:23 am, Aatu Koskensilta <aatu.koskensi...@xortec.fi> wrote: > Mathematicians don't think, or do not begin by thinking, the natural numbers > -- or the reals, or sets of sets of sets of reals, or what have you -- exist > and they can state and proves stuff about them? A very curious idea. Hardly. Factual examples do just beat you over the head. Sometimes the theory REALLY DOES get discovered BEFORE the model. Non-Euclidean geometry being the (really, really) obvious case in point. Seriously, the mere fact that THAT little episode in the history of consciousness EVERY happened REALLY should have just SHUT ALL of you UP, a hundred and fifty years ago! People thought GEOMETRY had an intended model TOO, at one time, you know. But (GEO-) it turns out that the EARTH'S surface really IS non-Euclidean! The whole concept of "the intended model" IS JUST BULLSHIT. I repeat, that is NOT EVEN OPEN to DEBATE. FACTUAL EXAMPLES are relevant. The parallel postulate IS FACTUALLY FALSE over the surface of the earth, even if you idealize the sphere. The theory was able to be investigated before that was even clarified. > > The completeness theorem is fundamentally a proof that first-order > > semantics SIMPLY DOESN'T EXIST. On Dec 19, 7:23 am, Aatu Koskensilta <aatu.koskensi...@xortec.fi> wrote: > Right. Just as the fundamental theorem of analysis establishes no-one knows > how to ride a bicycle. Abusive as usual. The completeness theorem is at least ABOUT first-order semantics. Where are the bicycles in analysis? > Abusive as usual. As usual. > The completeness theorem is at least ABOUT first-order semantics. > Where are the bicycles in analysis? Hiding behind the universal ordinal. The idea that "the completeness theorem is fundamentally a proof that first-order semantics SIMPLY DOESN'T EXIST" is entirely arbitrary and quite bizarre. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus On Dec 23, 1:58 pm, Aatu Koskensilta <aatu.koskensi...@xortec.fi> wrote: > > Where are the bicycles in analysis? > [quoted text clipped - 3 lines] > SIMPLY DOESN'T EXIST" is > entirely arbitrary and quite bizarre. This dismissal is entirely arbitrary. It has content zero. Would that it had measure zero as well. This dismissal does not analyze or assert anything. Not that anything really needs asserting. Everybody in the target audience ALREADY knows that the completeness theorem basically alleges that first-order |- "completely" covers first-order |= (that's why the theorem is named that). Given that everybody knows that, my "idea" was simply incontestable. It is no more arbitrary than "the sky is blue", and no more doubtful. This is a classic example of what I previously meant by disagreement with my allegedly idiosyncratic&bizarre opinons being, factually, neither coherent nor (therefore) possible. > This is a classic example of what I previously meant > by disagreement with my allegedly idiosyncratic&bizarre > opinons being, factually, neither coherent nor (therefore) > possible. Bah. We could with equal justification, and equally arbitrarily, claim that "the completeness theorem is fundamentally a proof that first-order syntactic deduction SIMPLY DOESN'T EXIST". SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus On Jan 1, 9:14 am, Aatu Koskensilta <aatu.koskensi...@xortec.fi> wrote: > Bah. We could with equal justification, and equally arbitrarily, claim that > "the completeness theorem is fundamentally a proof that first-order syntactic > deduction SIMPLY DOESN'T EXIST". Since the completeness theorem is, itself, LIKE ALL theorems, an exercise IN AND USING syntactic deduction that is NOT going to have "equal" justification. And none of it is arbitrary to begin with. You are just fulminating. Incompetently. On Jan 1, 9:14 am, Aatu Koskensilta <aatu.koskensi...@xortec.fi> wrote: > Bah. We could with equal justification, and equally arbitrarily, claim that > "the completeness theorem is fundamentally a proof that first-order syntactic > deduction SIMPLY DOESN'T EXIST". The theorem wasn't attempted in that direction. It was NOT like we had a PRIOR definition of logical consequence (over infinite domains, with finitary machinery) and were proving that we could re-cast it another way (model theoretically). The original Tarskian definition is model-theoretic. The question is whether any finitary syntactic system can "completely" emulate it. All of your allegation of absurdity here is founded on the superficial observation that proving an equivalence between two things "cannot" be proving that either of them doesn't exist. Nobody is impressed. You are basically flaunting an unwillingness (NObody in the target audience, especially not you, is actually stupid enough to flaunt inability) to apply the relevant metaphor. > Of course, once that regimented theory is on the table, It completely moots the relevance of any alleged prior model that may have inspired it. In the case of PA, this is more relevant than usual for the simple reason that the alleged prior model WAS INFINITE and therefore (arguably) not well-understood enough TO BEGIN WITH for ANYbody to have been able to WIN the argument over the question of whether he did or didn't KNOW WHAT HE WAS TALKING ABOUT (i.e. know what infinite model he originally had in mind). > Wanting maximal rigour > and absolute clarity, we reflect on our practices, and regiment our > informal mathematical language, and regiment chunks of our everyday > mathematics into nicely disciplined axiomatic systems. This is oversimplified and missing the point. A GREAT MANY DIFFERENT things could qualify as "nicely disciplined axiomatic systems". We are NOT talking about that huge total generality. We are only talking about ONE thing. WE are talking about classically-defined first-order languages over finitary signatures. > We construct, for example, the regimented theory > most of us call first-order PA. This is A COMPLETE LIE. *WE* do NOT do ANYthing. The first-order language of PA is a purely abstract formal thing with some simple known signature that ANYbody could've written down or stipulated WITHOUT CARING A FIG about what informal mathematical realm it might or might not have been "about". Mathematical practice IS PROVING stuff. In the first-order paradigm, proof is syntactic and INDEPENDENT of semantics. > This theory is as semantically contentful as the informal inchoate > theory we started off with, just better disciplined. It's simultaneously more AND less contentful. It's UNIVERSALLY contentful in that the theorems must be true of ALL models, both actual AND POSSIBLE, of the axioms. It is about a great many MORE semantic universes than may have motivated your initial investigation. My point is simply that the initial investigation is just irrelevant. The axioms are what matter. That is what is ACTUALLY being investigated. This no doubt seems to you like the inmates taking over the asylum; the axioms were CREATED *by* you to *serve* you in your quest for more TRUTH about the INTENDED model, in your opinion. In my version it winds up seeming like the axioms are the masters and you are doomed to slave away deriving their consequences. Yes, I suppose it is a lot more fun to imagine yourSELF and YOUR creativity in charge. >>> know that logic books occasionally talk like this, but this has always >>> seemed to me simply to be a misuse of the word "language". [quoted text clipped - 5 lines] > > I wasn't "arguing with a definition"; > but it is a somewhat misleading > definition, since uninterpreted syntactic systems in themselves convey > no messages, have no semantic content, You're right on this: it has no semantic content that we could care. > can't be used to communicate This is where you've gone wrong: we'd use the non-semantic formal language to formal theory, then try to mentally come up with a model (interpretation) where we could "see" some truths. Then we'd communicated the interpreted truths to others. It looks straight forward and simple, isn't it? > anything. And yet you might well suppose that it is ANALYTIC of the > ordinary notion of a language that it is a means for communication. [quoted text clipped - 12 lines] > supposed to be a *language* in which we can communicate some > mathematical truths". > Obviously, in your sense of formal language, > where formal languages have no content, a formal language cannot be > used as a vehicle to do mathematics. As mentioned, where you've gone wrong on this is the belief that "a formal language cannot be used as a vehicle to do mathematics". We've been doing it all the time, to varying degrees! > Fair enough. Such objects may be > interesting but not what I was talking about. I was talking about > contentful formalized languages, such as the languages of Principia, > or of first-order PA, or ZFC, meaningful languages in which we *can* > do more or less mathematics. > > In saying that the FOL with a given signature is unique (I take it you > > mean up to boring relabelling of predicates etc) suggests that you are [quoted text clipped - 9 lines] > Over here it is entirely normal to talk about a hierarchy of > formal languages (regular, context-free, etc.). Well, Peter is objecting to standard usage. You do that too! -- hz > > > In saying that the FOL with a given signature is unique (I take it you > > > mean up to boring relabelling of predicates etc) suggests that you are [quoted text clipped - 14 lines] > -- > hz Well, I'm not so sure that there *is* a standard usage of "language" among logicians. It seems to depend a bit, judging from the exchange between George and me, on whether you are coming from comp sci or philosophy. Not that the debate is, or ought to be, one about terminology. > Well, I'm not so sure that there *is* a standard usage of "language" > among logicians. But one difference between me and some other people is that *I* would *not* PRESUME to generalize over anything as broad as "among logicians". I'm NOT TRYING TO TALK about that. I'm talking about a standard usage of "language" that remains standard within a domain of discourse extending across ONE TEXTBOOK AND *NO* further, and it had better be a textbook ABOUT formal languages or theory of computing. I have no quarrel with everybody continuting to use "language" to mean something richer outside those textbooks. The question is, if you VOLUNTARILY limit yourself to such an impoverished conception of "language", how much CAN YOU STILL achieve? The answer is, with some reasonable axioms, quite a bit. If the axioms are as simple as PA, however, there are a lot of first-order truths about the naturals that you cannot achieve. You can achieve more of them, however, with an even SIMPLER first-order language, using first-order ZFC. My point is, the richness of what you can achieve is sufficiently great that nobody is, really, especially not in THIS room, ENTITLED to COMPLAIN that this sense of "language" is unnaturally or unreasonably impoverished, EVEN THOUGH IT IS. Some students actually do grow up to overcome the poverty of their beginnings. > > > I know that logic books occasionally talk like this, but this has always > > > seemed to me simply to be a misuse of the word "language". Herbzet defends: > Well, Peter is objecting to standard usage. You do that too! This analogy is misguided to put it charitably. Prof.Smith is actually calling the whole formal approach NON-standard. "Language" ALREADY HAD a meaning before formalists decided to make it a technical term. But that is ridiculous AS an objection. There are many words in the dictionary that list several incompatible meanings. If I were even TRYING to use language in the *same* usual standard sense that Prof.Smith is talking about, then he would have a point. But I really am not, and neither is CS as a field. The point is that the study of PURELY formal languages is a WELL-respected subfield of the field. We are not "misusing" the word "language" and we are not deviating from the non-CS "standard" use EITHER! We are talking about a DIFFERENT concept! We are *intentionally* restricting our attention to a PURELY formal extensionally-conceived collection! The fact that the word we are using for it is fraught with all these other "natural" connotations IS NOT EVEN RELEVANT. We are not "misusing" the word by proscribing those connotations OUT of consideration. We are rather simply using the word in a DIFFERENT narrow technical SENSE. "Chair" doesn't always mean a thing that you sit on with a seat and 4 legs. Sometimes it means "presiding officer" of a discussion meeting. To say that calling a formal language "a language" is misusing the word is ABUSIVE, frankly. IF we were actually trying to TRADE on or invoke or invite inference from ANY of the more "normal" attributes of the more normal definition of the word "language", then, yes, he would have a point, WE would be guilty of "misuse". BUT WE'RE NOT. WE ONLY mean what WE mean. We're ONLY talking about the formal version. On Dec 14, 11:02 am, tc...@lsa.umich.edu wrote: > Etchemendy is concerned with the right way to analyze the concept of logical > consequence, and is not claiming that what mathematicians regard as "truths" > are not really true. If you are going to cite the review then it wuold HELP if YOU would at least seem to have READ it! The first two paragraphs of the review point out that "Etchemendy intends his [refutation] to apply to "the standard semantic account", i.e., model-theoretic semantics.", and if he's right, "we must re- think the role played by model-theoretic semantics". The standard semantics really is model-theoretic and mathematicians AND EVERYBODY ELSE standardly DO understand truth as something that models and interpretations can ascribe to sentences-as-members- of- sets, i.e., sentences-as-theorems-of-theories. So your claim that he "is not claiming that what mathematicians regard as truths are not true" is hasty to put it charitably. The argument *I* have been having involves ONE case where mathematicians themselves are NOT adhering to the usual model-theoretic definition of truth to begin with -- the case of "interpreted languages". But the model-theoretic definition IS -- DESPITE that exceptional EXCEPTION by mathematicians -- USUAL. So if JE is attacking model-theoretic semantics then he certainly cannot avoid attacking model-theoretic truth right along with it. >So if JE is attacking model-theoretic semantics then he certainly cannot >avoid attacking model-theoretic truth right along with it. Sure. He's arguing that Tarski's approach is the wrong way to *analyze* logical consequence, and by extension, truth. But saying that someone's philosophical analysis of truth is incorrect doesn't mean that you disagree with the actual truth values of any particular statement (such as "there are infinitely many primes" or "PA is consistent"). SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences >But as for the Peano axioms, why *is* it restricted to >first order formulas? As opposed to what? Second-order >formulas? Does some kind of inconsistency result from >applying induction to higher-order formulas? There's no "need" to restrict to first-order formulas. There are a couple of reasons why logicians devote a lot of attention to the variant of the Peano axioms with induction restricted to first-order formulas. One reason is that first-order logic has some nice properties, such as the completeness theorem. Another reason is that the machinery of first-order logic can be set up with a minimum of conceptual prerequisites. In particular one need not refer to *sets* of integers. But from the point of view of a "working mathematician," not only is the restriction to first-order formulas far from mandatory---it's quite unnatural. The working mathematician is most interested in the ring of integers (I blur the distinction between integers and natural numbers here). PA does not single out the ring of integers---there are all those silly nonstandard models that it fails to exclude. For this reason, the working mathematician finds it much more natural to use a second-order induction axiom, because then one can prove that there is only one structure (up to isomorphism) satisfying them, namely the ring of integers. SignatureTim Chow tchow-at-alum-dot-mit-dot-edu The range of our projectiles---even ... the artillery---however great, will never exceed four of those miles of which as many thousand separate us from the center of the earth. ---Galileo, Dialogues Concerning Two New Sciences On 11 Dec, 18:58, tc...@lsa.umich.edu wrote: > But from the point of view of a "working mathematician," not only is the > restriction to first-order formulas far from mandatory---it's quite unnatural. [quoted text clipped - 5 lines] > then one can prove that there is only one structure (up to isomorphism) > satisfying them, namely the ring of integers. What Tim Chow says is basically right, of course. But there is also something to be added. In grasping full second-order PA, we need to get our heads around the idea not just of sets of numbers but the infinitary idea of arbitrary infinite sets of numbers. And you might reasonably suppose that that grasping *that* idea involves grasping more than is required to understand basic arithmetic. Arguably -- famously, Dan Isaacson has argued this, and I've indirectly defended this in print -- first order PA in fact is a "natural" theory as it captures just the truths that are accessible to a reasoner merely in virtue of their understanding of arithmetical ideas. To grasp other claims formulable in the language of first order arithmetic as true (e.g. to grasp as true PA Gödel's sentence, or to grasp as true the arithmetization of Goodstein's theorem) involves making use of "higher order" concepts, concepts that go beyond what are needed just to understand basic arithmetic. (Go to "other materials" at www.godelbook.net for a relevant short piece of mine.) > In grasping full second-order PA, we need to get our heads around the > idea not just of sets of numbers but the infinitary idea of arbitrary [quoted text clipped - 11 lines] > making use of "higher order" concepts, concepts that go beyond what > are needed just to understand basic arithmetic. Actually, to bring up an old point again, although it is perfectly standard and a handy way of talking, from a certain point of view it might be thought strange to call first order PA "arithmetical" and not use the same word for a second order arithmetic statement. In as far as "arithmetical" might be interpreted to mean "about numbers", some true properties of numbers require second order quantifiers to even state them, let alone prove them. Any set of natural numbers has a least element. That is second order and cannot be stated in first order form. You can talk about first order formulas, but then someone could come along and say "no, I really mean any set of numbers whatsoever has a least element". It seems to me hard to deny this property of the natural numbers, and hard to deny that it is as much "about numbers" as any statement possible could be. So in that sense, it is an arithmetical statement. I guess what I am trying to say is, you can go beyond the language of first order arithmetic but still be using perfectly "arithmetical" reasoning, if "arithmetical" is interpreted to mean "about numbers". As for whether, in order for humans to deduce certain first order arithmetic statements, it is not the case that there are first order arithmetic principles which are obvious or intuitive to humans and in principle allow us to do a proof of the theorem, but there are second or higher order principles that do the job, that may well be the case. Note that it is not meaningless to talk about a statement being or not being about numbers. Not every statement of mathematics is such a statement, for example the axiom of choice. I forgot to add, I don't think there is anything much "special" or "natural" about PA from the point of view of how PA is related to the mathematical objects we call "numbers". It is only special from the point of view of it's relation to the human mind. On Dec 11, 11:22 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > But as for the Peano axioms, why *is* it restricted to > first order formulas? abo got this right; I am just piling on. You have an antecedent failure on "it" above. > As opposed to what? Before there is PA, there is the prior question, "what language am I speaking"? "It" is restricted to first-order formulas because "it" IS A FIRST- ORDER LANGUAGE. "It" IS A *first*-order axiom-set. "It" is a FIRST-order TREATMENT OF THE WHOLE SUBJECT. So there is simply no question of an individual axiom "restricting" anything to first-order. First-order is simply the context we are in; it is the continent we inhabit, the atmosphere we breathe, the ocean in which we swim. That (first-order formulae) is all there is and there ain't no more, availability-wise. > Second-order formulas? No, first-order PREDICATES as first-order OBJECTS in a SECOND- order UNIVERSE. THAT is the alternative. You do NOT have to change ONE iota, jot, or tittle of the typography of the axiom-strings to invoke that alternative. ALL you have to say is that THE CONTEXT is now 2nd-order instead of 1st. Some people might say you also needed to add a universal 2nd-order quantifier Aphi at the beginning of the induction axiom but my point is, universal quantifiers are ALWAYS typographically optional; you can just define the language to have (allegedly) free variables (implicitly) universally quantified. > Does some kind of inconsistency result from > applying induction to higher-order formulas? Not for anything as simple as 2nd-order PA, certainly. Newberry says... >We can as well program a machine to say that PA is inconsistent or >that it does not know. Is it a pure accident that PA is consistent and [quoted text clipped - 6 lines] >"PA is consistent" would appear absurd to us? Is this what >"psychological reason" means? I'm saying that there is no reason to think that human mathematical reasoning surpasses that of any machine, and Godel's theorem in no way suggests otherwise. -- Daryl McCullough Ithaca, NY > I think you would agree that the consistency of PA is a hard fact not > just an emotional state or a similar psychological phenomenon. Of course it is, but why or whether you *believe* or agree with it is equally obviously psychological. The consistency of PA is equivalent to the question of whether a certain TM will or won't ever halt -- PA is consistent if this TM does not halt and inconsistent if it does. > means that no matter how long and in what order we keep generating > theorems we will never derive P & ~P. PA can be literally materialized > as a mechanical system, a machine. We are asking if tis machine can > ever produce P & ~P. No machine can answer that question. This is obviously false. One of two machines can answer this question. Either the machine that answers EVERY question "no" OR the machine that answers EVERY question "yes" can answer this question -- CORRECTLY. Another machine that generates theorems of ZFC can answer this question -- CORRECTLY. > We are able to answer it, hence it we surpass any machine. No, WE are NOT able to answer it, since after the ZFC-machine answers it "true", WE still have to wonder "but what if ZFC is inconsistent?" -- the ZFC-machine doesn't care. So it is more confident than we are. On Dec 8, 12:43 pm, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 10 lines] > we have experience with arithmetic, and all the axioms of > PA seem to be true, according to our experienc So we are convinced about the truth of Peano Arithmetic not because of a formal proof i.e. we surpass any machine. > >Contrary to your earlier claim, the following proof is not > >formalizable: [quoted text clipped - 13 lines] > Yes, S would be incomplete, but no, its consistency would > not be unknown. The cornerstone of TF's argument is that the consistency of ZFC is provable in ZFC + an axiom of infinity, which is no longer manifestly true. We hesitate as we go higher up in the chain of theories, hence - according to TF - we are not better than any machine because we cannot say about an arbitrary system that its Goedel formula is true. Yet he is absolutely sure that PA and ZFC are consistent. So if we are not sure if ZFC + an axiom of infinity are true the formal proof of ZFC's consistency may not tell the truth i.e. it does not have any cogency. Yet he is sure ZFC is consistent, but again not becuase of any formal proof. In fact his argumen proves that we do surpass any machine (if you reject 1 and 3.) Even if the truth of S were known and the truth of S_1 in which the consistency of S were proven, at some point we will reach a theory S_n which will not appear to be manifestly true. > What you can conclude, based on Godel's theorem, plus > some plausible assumptions about what human mathematicians [quoted text clipped - 8 lines] > not be able to become absolutely convinced of the consistency > of S. Are you know voting for option 1)? "Not absolutely convinced"? What does that mean? Are you saying that PA (ZFC) is probably consistent? How do you estimate the likelyhood that they are consistent? It makes sense to say that Goldbach conjecture is probably true, by which we mean that a proof will be eventually found. But what does it mean to be probably true if we know that a proof cannot exist? > -- > Daryl McCullough > Ithaca, NY > The cornerstone of TF's argument is that the consistency of ZFC is > provable in ZFC + an axiom of infinity, which is no longer manifestly > true. We hesitate as we go higher up in the chain of theories, hence - > according to TF - we are not better than any machine because we cannot > say about an arbitrary system that its Goedel formula is true. Yet he > is absolutely sure that PA and ZFC are consistent. Where did TF claim that he was "absolutely sure that ... ZFC [is] consistent"? I see no evidence that that was his view in the cite you posted earlier in the thread. SignatureAlan Smaill > > The cornerstone of TF's argument is that the consistency of ZFC is > > provable in ZFC + an axiom of infinity, which is no longer manifestly [quoted text clipped - 8 lines] > I see no evidence that that was his view in the cite you posted > earlier in the thread. He has an entire chapter in his book "Skepticism and Confidence", where he refutes the skeptics. "And given that the axioms of ZFC are so utterly compelling, so obviously true in the world of sets, we can do no better than to adopt these axioms as our starting point. Since the axioms are true, they are also consistent." [p.105] "if the axioms of ZFC are manifestly true, they are obviously consistent" [p.105] But it does not matter if TF is convinced that ZFC is consistent. I claim that there are only three possibilities: 1) We do not know if PA's Goedel sentence is true. So we do not know that the set of truth is productive. 2) The human mind surpasses any machine 3) There axists a formalization of arithmetic that can prove its own consistency Thus far I have not seen any convincing argument that we can reject all three. In fact based on what I have seen I am convinced that we cannot. Are you opting for 1)? You are a second convert. Daryl is now also leaning in that direction. It would make more sense than attempting to argue that we can reject all three, but it is quite a departure from what most logicians believe. Peter Smith will certainly differ. > But it does not matter if TF is convinced that ZFC is consistent. I > claim that there are only three possibilities: [quoted text clipped - 3 lines] > 3) There axists a formalization of arithmetic that can prove its own > consistency [I assume "PA" in 1) is a 'typo' for "ZFC"] Well, 3) is obviously true; any inconsistent formalization will do nicely. If you meant consistent formalization, you're wrong; one could for example hold that the human mind is equivalent in power to some finite extension of ZFC which proves the consistency of ZFC [for definiteness' sake, let's take the extra axiom to be Projective Determinacy]. Then 1) fails by hypothesis, 2) fails since the extension may be simulated by a machine, and 3) fails by Godel's Theorem. <berry@pop.networkusa.net> ha scritto >> But it does not matter if TF is convinced that ZFC is consistent. I >> claim that there are only three possibilities: [quoted text clipped - 12 lines] > definiteness' sake, let's take the extra axiom to be Projective > Determinacy]. You can't be really definite about it: if the human mind is equivalent to a system than it cannot ever know which system it is equivalent to (by Godel's theorem). > <be...@pop.networkusa.net> ha scritto > [quoted text clipped - 6 lines] > equivalent to a system than it cannot ever know which system > it is equivalent to (by Godel's theorem). Certainly, the equivalence cannot be established by a philosophical argument; but it might be established by some other means, such as an empirical argument. But actually I doubt that this system to which the human mind is equivalent can be naturally expressed in any language that humans can understand. So in particular, it can't be a finite extension of ZFC or any other known system. LordBeotian says... >You can't be really definite about it: if the human mind is equivalent to a >system than it cannot ever know which system it is equivalent to (by Godel's >theorem). Godel's theorem doesn't imply that. What it does imply is that *if* human mathematical reasoning is summed up by a recursively enumerable theory T, then we cannot know that T is consistent. I don't see any reason that we wouldn't be able to know that T is equivalent to our reasoning. -- Daryl McCullough Ithaca, NY "Daryl McCullough" <stevendaryl3016@yahoo.com> ha scritto >>You can't be really definite about it: if the human mind is equivalent to a >>system than it cannot ever know which system it is equivalent to (by [quoted text clipped - 6 lines] > I don't see any reason that we wouldn't be able to know that > T is equivalent to our reasoning. If I would know that T is equivalent to our reasoning I would think that T is consistent. LordBeotian says... >"Daryl McCullough" <stevendaryl3016@yahoo.com> ha scritto > [quoted text clipped - 11 lines] >If I would know that T is equivalent to our reasoning I would think that T is >consistent. But would you *know* that T is consistent? If so, how would you know that? -- Daryl McCullough Ithaca, NY "Daryl McCullough" <stevendaryl3016@yahoo.com> ha scritto >>>>You can't be really definite about it: if the human mind is equivalent to >>>>a [quoted text clipped - 13 lines] > > But would you *know* that T is consistent? If so, how would you know that? I certainly would have a justified true belief, and it also would seem to be knowledge to me, if it is actually knowledge or not (and which justified true beliefs are actually knowledge) is a hard philosophical problem that I am not able to address here. LordBeotian says... >"Daryl McCullough" <stevendaryl3016@yahoo.com> ha scritto >>>If I would know that T is equivalent to our reasoning I would think that T >>>is [quoted text clipped - 3 lines] > >I certainly would have a justified true belief, Why would it be justified? >and it also would seem to be >knowledge to me, if it is actually knowledge or not (and which justified true >beliefs are actually knowledge) is a hard philosophical problem that I am not >able to address here. I'm not really trying to quibble about what "knowledge" means. If you want it to mean something short of proof, that's fine. But it seems to me that there are *two* claims that could be justified beliefs: 1. The claim that T is actually *your* theory (that is, T contains all the statements of arithmetic that you could come to know were true). More specifically, if Phi is any statement of arithmetic, then you can come to know Phi if and only if T proves Phi. 2. The claim that T is consistent. Both claims could have substantial empirical support. To support the claim that T is consistent, you could spend a few decades attempting and failing to derive a contradiction from T in all the known ways: Russell paradox, Burali-Forti paradox, etc. To support the claim that T captures your theory, you could look at the program by which T is produced, to see that it really captures the way your brain works. You could try (unsuccessfully) to come up with some new mathematical claim that you believe but that T is incapable of proving. So you could have justified belief in both claims. But could you be said to *know* the truth of both claims? It depends on your standard of what knowledge is. But if you use the *same* standard of knowledge as is implicit in claim 1, then you are in trouble if you claim to know both statements. If you claim to know 2, that T is consistent, then it would follow that the arithmetic statement Con(T) is true. So you would know that Con(T) is true. But if you also know 1, then it would follow that T can prove Con(T), which would imply that T is inconsistent. So you can't consistently know both 1 and 2. I don't see any problem with knowing 1 and being uncertain about 2, but you seem to be claiming that knowing 1 implies knowing 2. -- Daryl McCullough Ithaca, NY > If I would know that T is equivalent to our reasoning I would think that T is > consistent. Not necessarily; you (obviously) don't know that your own reasoning is consistent. "george" <greeneg@cs.unc.edu> ha scritto >> If I would know that T is equivalent to our reasoning I would think that T >> is >> consistent. > > Not necessarily; you (obviously) don't know that your own reasoning > is consistent. What do I know? > What do I know? Well, that depends on whether your reasoning is consistent. Just because you can't prove it is doesn't mean it isn't. On Dec 8, 11:15 pm, be...@pop.networkusa.net wrote: > > But it does not matter if TF is convinced that ZFC is consistent. I > > claim that there are only three possibilities: [quoted text clipped - 14 lines] > extension may be simulated by a machine, and 3) fails by Godel's > Theorem. Goedel disproved 3) for classical logic with Peano Axioms, not for any formalization of arithmetic. > On Dec 8, 11:15 pm, be...@pop.networkusa.net wrote: > [quoted text clipped - 19 lines] > Goedel disproved 3) for classical logic with Peano Axioms, not > for any formalization of arithmetic. No; you need look no farther than the title of his paper to see what system he was actually considering. Of course he also generalized his results to "related systems", which would include any finite extension of ZF, among other things. [It can be generalized even further than Godel did, of course]. On Dec 9, 10:41 am, be...@pop.networkusa.net wrote: > > On Dec 8, 11:15 pm, be...@pop.networkusa.net wrote: > [quoted text clipped - 25 lines] > of ZF, among other things. [It can be generalized even further than > Godel did, of course] But you need to establish that it generalizes to ALL conceivable formalizations of arithmetic. .- Hide quoted text - > - Show quoted text - >> > The cornerstone of TF's argument is that the consistency of ZFC is >> > provable in ZFC + an axiom of infinity, which is no longer manifestly [quoted text clipped - 11 lines] > He has an entire chapter in his book "Skepticism and Confidence", > where he refutes the skeptics. He has more than one book; do you mean in "Gödel's Theorem: An Incomplete Guide to its Use and Abuse"? > "And given that the axioms of ZFC are so utterly compelling, so > obviously true in the world of sets, we can do no better than to adopt > these axioms as our starting point. Since the axioms are true, they > are also consistent." [p.105] I'll look at the context before commenting, since that is very different from his attitude to ZFC in "Inexhausibility". > "if the axioms of ZFC are manifestly true, they are obviously > consistent" [p.105] Clearly that is a hypothetical statement, which does not commit him to the antecedent. > But it does not matter if TF is convinced that ZFC is consistent. I > claim that there are only three possibilities: > 1) We do not know if PA's Goedel sentence is true. So we do not know > that the set of truth is productive. That's a bunch of stuff you have already bundled up -- the 'so' part is not a logical consequence of the the first sentence, for a start. > 2) The human mind surpasses any machine > 3) There axists a formalization of arithmetic that can prove its own [quoted text clipped - 5 lines] > > Are you opting for 1)? Maybe you can clarify what you are asking about: that we do not know that PA's Goedel sentence is true; that we do not know if the set of truths is productive; or that the second follows from the first? > You are a second convert. Daryl is now also > leaning in that direction. It would make more sense than attempting to > argue that we can reject all three, but it is quite a departure from > what most logicians believe. Peter Smith will certainly differ. SignatureAlan Smaill > >> > The cornerstone of TF's argument is that the consistency of ZFC is > >> > provable in ZFC + an axiom of infinity, which is no longer manifestly [quoted text clipped - 14 lines] > He has more than one book; do you mean in > "Gödel's Theorem: An Incomplete Guide to its Use and Abuse"? Yes, I mean this one. > > "And given that the axioms of ZFC are so utterly compelling, so > > obviously true in the world of sets, we can do no better than to adopt [quoted text clipped - 33 lines] > that we do not know if the set of truths is productive; > or that the second follows from the first? OK, forget about the second part for now. Let's confine ourselves to 1) We do not know if PA's Goedel sentence is true. > > You are a second convert. Daryl is now also > > leaning in that direction. It would make more sense than attempting to [quoted text clipped - 5 lines] > > - Show quoted text - >> > The cornerstone of TF's argument is that the consistency of ZFC is >> > provable in ZFC + an axiom of infinity, which is no longer manifestly [quoted text clipped - 11 lines] > He has an entire chapter in his book "Skepticism and Confidence", > where he refutes the skeptics. Having dug out the book in question, I now have the context of the following. > "And given that the axioms of ZFC are so utterly compelling, so > obviously true in the world of sets, we can do no better than to adopt > these axioms as our starting point. Since the axioms are true, they > are also consistent." [p.105] The very next sentence says: "Again, the point at issue is not whether such a view of the axioms of ZFC is justified, but whether it makes good sense to appeal to the incompleteness theorem in criticism of it." The context is exactly a critique of the argument that suggests that *if* someone has certain knowledge of the truth of the axioms of a system *then* they will run into trouble from the incompleteness theorem. So the view expressed are *hypotheical*, not TF's at all. And having set up the context clearly, TF reminds the reader of the context afterwards. > "if the axioms of ZFC are manifestly true, they are obviously > consistent" [p.105] As I said before, this is a conditional statement. Why on earth do you think it expresses a commitment to the antecedent being true? At this rate someone will have to give us "The uses and abuses of TF's writings". (And, while I'm here, he does not refute scepticism in this chapter.) SignatureAlan Smaill > >> > The cornerstone of TF's argument is that the consistency of ZFC is > >> > provable in ZFC + an axiom of infinity, which is no longer manifestly [quoted text clipped - 19 lines] > > these axioms as our starting point. Since the axioms are true, they > > are also consistent." [p.105] This is a clear affirmative statement. He believes that the axioms are manifestly true. > The very next sentence says: > [quoted text clipped - 6 lines] > of a system *then* they will run into trouble from the incompleteness > theorem. He is aruing that Goedel's theorem cannot be used to doubt the truth of ZFC.It is not mutually exclusive with the belief that the axioms of ZFC ARE manifestly true. Basically he is saying that EVEN IF you do not believe that the axioms of ZFC are manifestly true you cannot use Goedel's theorem to doubt ZFC. Read it more carefully. His book is not exactly clear in the sense that he jumps from aruing one point to another without making it explicitly clear. There is no one single thread of reasoning. > So the view expressed are *hypotheical*, not TF's at all. > And having set up the context clearly, TF reminds the reader [quoted text clipped - 11 lines] > > (And, while I'm here, he does not refute scepticism in this chapter.) What do you think he is doing? > -- > Alan Smaill- Hide quoted text - > > - Show quoted text - >> >> > The cornerstone of TF's argument is that the consistency of ZFC is >> >> > provable in ZFC + an axiom of infinity, which is no longer manifestly [quoted text clipped - 22 lines] > This is a clear affirmative statement. He believes that the axioms are > manifestly true. Only if you ignore the context!! In context, it is completely clear that he envisages a *what if* scenario, and is not presenting his own opinion. >> The very next sentence says: >> [quoted text clipped - 9 lines] > He is aruing that Goedel's theorem cannot be used to doubt the truth > of ZFC. Yes (in a particular context) -- that doesn't mean he *is* arguing that ZFC is true, which is your claim. > It is not mutually exclusive with the belief that the axioms of > ZFC ARE manifestly true. Agreed; but, contrary to your claim, he does *not* make the claim that they are true, which is a whole different situation. > Basically he is saying that EVEN IF you do > not believe that the axioms of ZFC are manifestly true you cannot use > Goedel's theorem to doubt ZFC. More accurately, he's saying that Goedel's theorem gives no help to a sceptic arguing against someone who thinks that the axioms of ZFC are clearly true. > Read it more carefully. I have read it carefully; there are simnply no grounds there for saying that *TF* believed that the axioms of ZFC are clearly true, as you suggest. > His book is not exactly clear in the sense > that he jumps from aruing one point to another without making it > explicitly clear. There is no one single thread of reasoning. He works hard to make clear what is at issue at any moment, and it's clear here that the statement you quoted is in the context of a temporary assumption that someone takes the axioms to be manifestly true, in order for TF to see what the consequences are. >> So the view expressed are *hypothetical*, not TF's at all. >> And having set up the context clearly, TF reminds the reader [quoted text clipped - 13 lines] > > What do you think he is doing? He's saying that the sceptics are not justified in using Goedel's theorem as evidence for their position, and that scepticism can only be justified on other grounds -- he's not saying that there is no justification for scepticism. SignatureAlan Smaill > I have read it carefully; > there are simnply no grounds there for saying that *TF* believed that > the axioms of ZFC are clearly true, as you suggest. OF course there are. Even worse, there are grounds for thinking that TF endorses the position that "manifestly true" is even meaningful, which is, in the context of formal languages, stupid. > > His book is not exactly clear in the sense > > that he jumps from aruing one point to another without making it > > explicitly clear. There is no one single thread of reasoning. I'm with the Newbie on this one. > He works hard to make clear what is at issue at any moment, sh.t. > and it's clear here that the statement you quoted is in the > context of a temporary assumption that someone takes > the axioms to be manifestly true, "taking the axioms to be manifestly true" IS INCOHERENT. For TF to EVEN do this HYPOTHETICALLY is for him to ASSERT that it is POSSIBLE/coherent TO do this. WHICH IT ISN'T. This is all alluding to a real world of sets, which is EQUALLY phlogistonic. It's all just silly. Once you KNOW that truth for these KINDS of sentences COMES FROM MODELS, you canNOT put the genie BACK in the bottle! Not unless you attack Tarskian interpretational semantics generally. All these people talking about "an interpreted language" ARE TALKING OUT THEIR a.s. Very much including TF, hypothetically OR OTHERwise. Even ENTERTAINING as OPPOSED to DISMISSING that hypothesis is ALREADY making a serious error. > in order for TF to see what the consequences are. You CANNOT have CONSEQUENCES until AFTER you have A LOGIC! But the nature of logic is precisely what is under dispute here! Moving the inquiry up to natural language does NOT help. You would need a PRIOR answer to "what is truth?", and in natural language, that is MAXIMALLY hard because you can say "This sentence is NOT true." > >> So the view expressed are *hypothetical*, not TF's at all. > >> And having set up the context clearly, TF reminds the reader > >> of the context afterwards. > > >> > "if the axioms of ZFC are manifestly true, they are obviously > >> > consistent" [p.105] This is obviously bullshit. If they were "manifestly" ANYthing then EVERYBODY WOULD OBVIOUSLY KNOW IT already. In general, PEOPLE DON'T. He might as well be inquiring about what would be happening if the moon were made of green cheese. SINCE IT OBSERVABLY ISN'T.... > >> As I said before, this is a conditional statement. > >> Why on earth do you think it expresses a commitment > >> to the antecedent being true? Because it contains "manifestly". Maybe you should look it up. > >> At this rate someone will have to give us "The uses and abuses > >> of TF's writings". You're off to a good start. But when something is just wrongheaded to begin with, surely letting it rest in peace is preferable. > He's saying that the sceptics are not justified in using Goedel's > theorem as evidence for their position, Of course they are. Something that is false in a model OBVIOUSLY CANNOT be "manifestly" true. > and that scepticism > can only be justified on other grounds -- he's not saying > that there is no justification for scepticism. Well, that would certainly be an improvement, but the underlying language (TF's language, not the underlying language of 1st-order ZFC) makes it impossible to get the assumptions right. Once you concede the possibility of "a real universe of sets", random BS ensues just as surely as if you had conceded the possibility of pigs flying or P&~P. TF and a whole bunch of other people have always taken it as clear that you can view the formal language as having a preferred interpretation. That is original sin in the context of this issue. >> I have read it carefully; >> there are simnply no grounds there for saying that *TF* believed that >> the axioms of ZFC are clearly true, as you suggest. > > OF course there are. Then please tell me where you find these grounds, because they are not to be found in the passage cited. > Even worse, there are grounds for thinking that TF endorses > the position that "manifestly true" is even meaningful, which > is, in the context of formal languages, stupid. He is not talking about formal languages here; he is assuming, for the purposes of discussion, that the someone adopts the position that the axioms are true statements about the (assumed) world of sets. >> > His book is not exactly clear in the sense >> > that he jumps from aruing one point to another without making it >> > explicitly clear. There is no one single thread of reasoning. > > I'm with the Newbie on this one. That rather depends who the Newbie is ... >> He works hard to make clear what is at issue at any moment, > sh.t. your mileage varies, we knew that. >> and it's clear here that the statement you quoted is in the >> context of a temporary assumption that someone takes [quoted text clipped - 4 lines] > ASSERT that it is POSSIBLE/coherent TO do this. > WHICH IT ISN'T. You are welcome to develop your own ideas on this, of course. The point of my post was simply to say that I find no evidence that TF himself thought that the axioms of ZF are manifestly true. If you have evidence that suggests otherwise, I'd welcome a view of that evidence. >> >> So the view expressed are *hypothetical*, not TF's at all. >> >> And having set up the context clearly, TF reminds the reader [quoted text clipped - 6 lines] > If they were "manifestly" ANYthing then EVERYBODY WOULD > OBVIOUSLY KNOW IT already. So the *conditional* statement is true, in your view, as an ex falso quodlibet. >> >> As I said before, this is a conditional statement. >> >> Why on earth do you think it expresses a commitment >> >> to the antecedent being true? > > Because it contains "manifestly". > Maybe you should look it up. Come on, it's in the antecedent of a conditional! >> >> At this rate someone will have to give us "The uses and abuses >> >> of TF's writings". > > You're off to a good start. you're welcome. > But when something is just wrongheaded to begin with, > surely letting it rest in peace is preferable. [quoted text clipped - 5 lines] > Something that is false in a model OBVIOUSLY > CANNOT be "manifestly" true. You are welcome to your own opinion; but this is an accurate report of TF's views in that chapter. If you dispute that this was *his* opinion, by all means explain why you think I misrepresent him. >> and that scepticism >> can only be justified on other grounds -- he's not saying >> that there is no justification for scepticism. SignatureAlan Smaill > >> there are simnply no grounds there for saying that *TF* believed that > >> the axioms of ZFC are clearly true, as you suggest. [quoted text clipped - 3 lines] > Then please tell me where you find these grounds, > because they are not to be found in the passage cited. OF course they are. Besides, the issue isn't even "truth" to begin with. The issue is TF's insistence on treating "truth" like it is coherent or reasonably understood. > > Even worse, there are grounds for thinking that TF endorses > > the position that "manifestly true" is even meaningful, which > > is, in the context of formal languages, stupid. > > He is not talking about formal languages here; I can't help it if you can read the book carefully and closely and still lie about it. The axioms ARE WRITTEN IN a formal language. He therefore CANNOT avoid being ABOUT formal languages. That IS his subject matter. That IS SO TO what he was talking about. If you are actually reading the book and still disputing this then there is little hope of an actual discussion. > he is assuming, for the > purposes of discussion, that the someone adopts the position that the > axioms are true statements This DOES NOT CHANGE the fact that those statements ARE WRITTEN IN A FORMAL LANGUAGE. > about the (assumed) world of sets. But assuming the existence of any such world is, I repeat, STUPID. If anyone tries to assume THAT then the simple refutation is "what is a set? I don't get these "sets" you're trying to talk about." ALL anyone CAN do by way of replying to that is GIVE A FORMAL LANGUAGE, which IS going to have MULTIPLE interpretations. That is just the end of it. >> >> there are simnply no grounds there for saying that *TF* believed that >> >> the axioms of ZFC are clearly true, as you suggest. [quoted text clipped - 5 lines] > > OF course they are. repetition does not add to your case. > Besides, the issue isn't even "truth" to begin with. > The issue is TF's insistence on treating "truth" like it is > coherent or reasonably understood. My issue is mundanely with the claim that *this passage* gives grounds for saying that TF claims that the axioms of ZF are clearly true. If your complaint is that he treats the notion as coherent, then I agree he does so, but that doesn't mean he (here) treats the position as true. >> > Even worse, there are grounds for thinking that TF endorses >> > the position that "manifestly true" is even meaningful, which [quoted text clipped - 5 lines] > still lie about it. The axioms ARE WRITTEN IN a formal language. > He therefore CANNOT avoid being ABOUT formal languages. I took your assertion to mean that the topic at hand is uninterpreted formal languages considered purely syntactically. That is not what is going on. The more important point is that TF is here arguing *hypothetically* -- what follows **if** someone takes this position? > That IS his subject matter. That IS SO TO what he was talking > about. If you are actually reading the book and still disputing > this then there is little hope of an actual discussion. Formal languages are involved, but not *just* formal languages in (what I take to be) your sense. >> he is assuming, for the >> purposes of discussion, that the someone adopts the position that the >> axioms are true statements > > This DOES NOT CHANGE the fact that those statements > ARE WRITTEN IN A FORMAL LANGUAGE. yes, they are. >> about the (assumed) world of sets. > > But assuming the existence of any such world is, I repeat, STUPID. It's consistent with everything on p106 of "G's theorem" that TF agrees with you that the existence of any such world is "STUPID". From the page in question: ".. we need to distinguish between two things: what degree of skepticism or confidence regarding mathematical axioms or methods or reasoning is justifiable or reasonable, and what bearing Gödel's theorem has on the matter. Perhaps we take a dim view of the claim that we know with absolute certainty the truth of, say, the axioms of ZFC, but how can we use Gödel's theorem to criticise this claim?" > If anyone tries to assume THAT then the simple refutation is > "what is a set? I don't get these "sets" you're trying to talk > about." > ALL anyone CAN do by way of replying to that is GIVE A FORMAL > LANGUAGE, which IS going to have MULTIPLE interpretations. Or maybe someone might try to bring Gödel's theorem to bear. That is what TF is on about here. > That is just the end of it. SignatureAlan Smaill > It's consistent with everything on p106 of "G's theorem" > that TF agrees with you that the existence of any such > world is "STUPID". Well, you have the page and I don't, but I still doubt you here. "any such world" is ambiguous. I am trying to deny 1 world and he is trying to deny another, I expect. > From the page in question: > > ".. we need to distinguish between two things: > what degree of skepticism or confidence regarding mathematical > axioms or methods or reasoning is justifiable or reasonable, Absolutely NO degree, OBVIOUSLY. That he is even entertaining this is, well, tragic & typical; but what is even MORE of both of the above is that THAT issue winds up getting CONFLATED with the SEPARATE issue of TRUTH. > and what bearing Gödel's theorem has on the matter. Perhaps > we take a dim view of the claim that we know with absolute > certainty the truth of, say, the axioms of ZFC, but how > can we use Gödel's theorem to criticise this claim?" Via the model existence theorem, obviously. This is completely easy and straightforward; indeed it is a matter almost purely of definition. As you are presenting his argument here it is worse than incoherent -- it is just the opposite of sound. > > ALL anyone CAN do by way of replying to that is GIVE A FORMAL > > LANGUAGE, which IS going to have MULTIPLE interpretations. > > Or maybe someone might try to bring Gödel's theorem to bear. That is NOT an ALTERNATIVE, dangit! Giving a formal language with multiple interpretations *constitutes* bringing Godel's theorem to bear! >> It's consistent with everything on p106 of "G's theorem" >> that TF agrees with you that the existence of any such >> world is "STUPID". > > Well, you have the page and I don't, I wondered why you were so convinced about what TF was claiming. > but I still doubt you here. "any such world" is ambiguous. > I am trying to deny 1 world and he is trying to deny another, > I expect. maybe so, but what's written is still consistent with what I take your opinion to be. >> From the page in question: >> [quoted text clipped - 3 lines] > > Absolutely NO degree, OBVIOUSLY. he makes the distinction just to say that this aspect is not at issue here. > That he is even entertaining this is, well, tragic & typical; but what > is [quoted text clipped - 11 lines] > his argument here it is worse than incoherent -- it is just > the opposite of sound. I'm not claiming that TF is right here; what I'm claiming is that nothing *here* commits him to the view that the axioms of ZFC are true. That was Newberry's claim, and that's the issue as far as I'm concerned. SignatureAlan Smaill > The point of my post was simply to say that I find no evidence > that TF himself thought that the axioms of ZF are manifestly true. > > If you have evidence that suggests otherwise, I'd welcome > a view of that evidence. You will find in the archives several posts in which Torkel explicitly says he finds the axioms of set theory manifestly true. That is of course entirely irrelevant to the point he makes in the quoted passage. SignatureAatu Koskensilta (aatu.koskensilta@xortec.fi) "Wovon man nicht sprechen kann, daruber muss man schweigen" - Ludwig Wittgenstein, Tractatus Logico-Philosophicus > The cornerstone of TF's argument is that the consistency of ZFC is > provable in ZFC + an axiom of infinity, ZFC *already* has "an" axiom of infinit TO START with. If you want to add another axiom then you are going to have to say *more* about it than merely that is "an" axiom "of" infnity. > which is no longer manifestly true. NOTHING is manifestly true except validities. EVERYthing else (except contradictions) is true IN SOME MODELS (or theories) and FALSE IN OTHERS. Your first mission is to prove that YOU understood what TF meant by "manifestly" and "of infinity". On Dec 8, 12:43 pm, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 7 lines] > That's a pychological question, not a mathematical question. > It just shows that certainty does not always come from proof. Sounds like you have just proved the human mind surpasses any machine ... > We are convinced about the truth of Peano Arithmetic because > we have experience with arithmetic, and all the axioms of > PA seem to be true, according to our experienc ... unless you claim that mathematics is empirical. That would be quite extraordinary. Nowdays it is fashionable to maintain a noble silence about the status of the propositions of mathematics (synthetic a posteriori, synthetic a priori, analytic) yet to claim that numbers are Platonic entities, which pre-exist in the outer space, and whose properties our formal systems only inadequately capture. So to take a position on the status of the propositions of mathematics is extraordinary. > >Contrary to your earlier claim, the following proof is not > >formalizable: [quoted text clipped - 30 lines] > Daryl McCullough > Ithaca, NY > Nowdays it is fashionable to maintain a noble > silence about the status of the propositions of mathematics Really? Fashionable in what quarters?? My shelves are weighed down with books on varieties of structuralism, fictionalism, neo- logicism, .... I can't see any sign of noble silence at all, but rather a lot of very vigorous debate. >yet to claim that numbers > are Platonic entities, which pre-exist in the outer space Really?? I can't think of *any* serious contemporary philosopher of mathematics who thinks that numbers exist or pre-exist in outer space. > > Nowdays it is fashionable to maintain a noble > > silence about the status of the propositions of mathematics [quoted text clipped - 3 lines] > logicism, .... I can't see any sign of noble silence at all, but > rather a lot of very vigorous debate. Do these fols have anything to say about synthetic/analytic? > >yet to claim that numbers > > are Platonic entities, which pre-exist in the outer space > > Really?? I can't think of *any* serious contemporary philosopher of > mathematics who thinks that numbers exist or pre-exist in outer space. TF frequently refers to numbers as if they existed before we defined them. Come to think of it, the idea that propositions can be true even though that they are not derivable kind of implies Platonism. On second thought I am not surprised that people think that way (i.e. Platonic.) That brings about the question: if numbers are Platonic entities, can mathematics possibly be analytic, and in either case how can machines comprehend Platonic entities (let aone synthetic a priori truth)? > > > Nowdays it is fashionable to maintain a noble > > > silence about the status of the propositions of mathematics [quoted text clipped - 5 lines] > > Do these fols have anything to say about synthetic/analytic? Some do, some would have Quinean reservations about the very synthetic/ analytic distinction. > > >yet to claim that numbers > > > are Platonic entities, which pre-exist in the outer space [quoted text clipped - 4 lines] > TF frequently refers to numbers as if they existed before we defined > them. But that isn't to say that they "exist or pre-exist in outer space" (or in any other location). > > Nowdays it is fashionable to maintain a noble > > silence about the status of the propositions of mathematics [quoted text clipped - 3 lines] > logicism, .... I can't see any sign of noble silence at all, but > rather a lot of very vigorous debate. Whaddya call it when it is asserted that mathematical existence is logically possible existence? -- hz Newberry says... >On Dec 8, 12:43 pm, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: [quoted text clipped - 12 lines] >Sounds like you have just proved the human mind surpasses any >machine ... How in the world did you come to that conclusion? To say that a human mind surpasses any machine says that there is something that a human can do that no machine can do. What is an example of such a thing? Believing in the consistency of PA? You can program a machine to do that. How is believing something without proof evidence that humans surpass machines? It's *easy* to program something that believes without proof. >> We are convinced about the truth of Peano Arithmetic because >> we have experience with arithmetic, and all the axioms of >> PA seem to be true, according to our experienc > >... unless you claim that mathematics is empirical. The fact that PA is consistent is certainly empirical. There are nonempirical ways to demonstrate its consistency, but they rely on other mathematical principles which we only accept for empirical reasons. There is *no* reason to believe that the way that humans learn mathematics could not be programmed into a machine. We don't know how to do it, but that isn't the issue. The issue is whether Godel's theorem implies that it is impossible to program a machine to do mathematics at the level of humans. Godel's theorem implies no such thing. -- Daryl McCullough Ithaca, NY On Dec 11, 6:04 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 21 lines] > do. What is an example of such a thing? Believing in the > consistency of PA? You can program a machine to do that. Either our belief that PA is sound has nothing to do with any facts about PA or it does. In the first case we could have just as well be programmed to believe that the consistency of PA was absurd. In the second case you need to explain how we derived the the conclusion that PA is consistent. > How is believing something without proof evidence that > humans surpass machines? It's *easy* to program something [quoted text clipped - 24 lines] > > - Show quoted text - Newberry says... >On Dec 11, 6:04 am, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >> To say that a human mind surpasses any machine says that >> there is something that a human can do that no machine can [quoted text clipped - 3 lines] >Either our belief that PA is sound has nothing to do with any facts >about PA or it does. Well of course it has to do with facts about PA. If, for example, PA were provably inconsistent, then we certainly wouldn't believe it were sound. >In the second case you need to explain how we derived the the >conclusion that PA is consistent. Why do I need to explain that? I didn't say that the machine would necessarily use the same mechanisms as a human, only that the result would be the same. Of course a machine will not use the same mechanisms as a human. Humans reason using blood and neurons and electrochemical reactions. I assumed by "the human mind surpasses any machine" you meant that a human mind can do something that a machine cannot, not that a human mind uses different mechanisms. If you are only claiming the latter, then it's not really a matter of the human *surpassing* the machine, as just being *different*. If I claim that I can jump higher than you can, and you claim I'm wrong, you're under no obligation to explain the mechanism that I use to jump. You just need to jump as high as I do. -- Daryl McCullough Ithaca, NY On Dec 11, 7:54 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 24 lines] > that a human mind can do something that a machine cannot, not > that a human mind uses different mechanisms. I think it is pretty clear what (not only) I meant. By "machine" we mean we mean "computable" e.g. as defined by Turing. A Turing machine can indeed be a realized a physical machine (modulo size restrictions.) Human can conclude that PA is consistent, which is not computable. (OK, it can be proved in ZFC, blah, blah, blah, let's not go around in circles, we do not know if ZFC is consistent so the formal proof amounts to nothing, but humans are certain that PA is consistent.) If you are only > claiming the latter, then it's not really a matter of the human > *surpassing* the machine, as just being *different*. [quoted text clipped - 6 lines] > Daryl McCullough > Ithaca, NY Newberry says... >I think it is pretty clear what (not only) I meant. By "machine" we >mean we mean "computable" e.g. as defined by Turing. A Turing machine >can indeed be a realized a physical machine (modulo size >restrictions.) Human can conclude that PA is consistent, which is not >computable. Do you know what "computable" means? Something is computable if there is a computer program that can reliable give the right answer. "Is PA consistent?" is perfectly computable. I can program write a program that gives the correct answer, "yes". >(OK, it can be proved in ZFC, blah, blah, blah, So you know that what you were saying was nonsense, and you said it anyway. That's bizarre. There is no sense in which "PA is consistent" is uncomputable. What you mean is that there is no *proof* of the consistency of PA that doesn't use equally (or more) dubious principles. That's probably true. So what? That means that humans are "programmed" to believe something like arithmetic even without proof. We can program a computer to believe something without proof. That's what an *axiom* is. Certain things you believe because they are "built-in". -- Daryl McCullough Ithaca, NY On Dec 12, 6:46 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > Newberry says... > [quoted text clipped - 13 lines] > So you know that what you were saying was nonsense, and you > said it anyway. That's bizarre. We already went throug this circe 15 times. a) The consitency of PA is provable b) But the proof does not habe any cogency c) Since we know that PA is consistent we surpass any machine d) But we do not the consistency of PA is provable etc. etc. > There is no sense in which "PA is consistent" is uncomputable. > [quoted text clipped - 5 lines] > proof. That's what an *axiom* is. Certain things you believe > because they are "built-in". We went through this one as well. We could have just as well been programmed to believe tha the consistency of PA was absurd. I think most sensible position would be to say that we are programmed as ZFC + plus some strong axiom of infinity. This is probably what TF meant. There is a problem with this too. Newberry says... >On Dec 12, 6:46 am, stevendaryl3...@yahoo.com (Daryl McCullough) >wrote: >We already went throug this circe 15 times. Yes, I know. I keep thinking that you can be convinced about how ridiculous your argument. It's complete nonsense. That's why I asked you to put it in the form of a syllogism, so you can see that your conclusions *don't* follow from your hypotheses. >a) The consistency of PA is provable >b) But the proof does not have any cogency I don't grant that at all. Why not say what is actually the case: "The proof of the consistency of PA (or *any* theory, for that matter) requires principles that go beyond PA". To conclude that that means that the proof has no cogency is just silly. >c) Since we know that PA is consistent we surpass any machine That's a non sequitur. You haven't demonstrated that no machine can "know" that PA is consistent, because you haven't defined the word "to know". Define it *first*. *Then* demonstrate that humans know that PA is consistent *using* this definition. *Then* demonstrate, using the *same* definition, that no computer program can know that PA is consistent. So there are at least three gaps in your argument. (1) You haven't defined what it means to "know" a mathematical fact. (2) You haven't demonstrated that humans know that PA is consistent using that definition. (3) You haven't demonstrated that machines cannot know that PA is consistent using that definition. There are several candidate definitions of "know" under which humans can be said to know that PA is consistent. There are other candidate definitions of "know" under which nobody ever knows anything. You are applying one definition to humans, and a *different* definition to machines. You have to *pick* one, and show that humans know PA is consistent by that definition, and that machines cannot know it, by that same definition. -- Daryl McCullough Ithaca, NY On Dec 12, 9:46 am, stevendaryl3...@yahoo.com (Daryl McCullough) wrote: > There is no sense in which "PA is consistent" is uncomputable. Of course there is. > What you mean is that there is no *proof* of the consistency > of PA that doesn't use equally (or more) dubious principles. So, you concede, there is THAT sense. > Hold on, hold on. You've rather ripped this out of context from Dan's > lecture notes. He two paragraphs earlier introduces the assumption > that we are dealing with systems of arithmetic S which are *sound* > (and hence consistent). Come ON. That is a lame dodge on YOUR part. HE is the one who changed the context by drawing THE DIRECT ANALOGY WITH NON-Euclidean geometry! NObody deprecates non-Euclidean geometries as "unsound"! That parlance in the general community IS INDEFENSIBLE. The relevant discovery is that "unsound" models of PA EVEN *can* Exist at all! Since the only reason GIT is important is that it proves THAT, for ANYbody to be trying to confine the discussion to "sound" systems IS IDIOTIC. The whole import of GIT is that PA+~Con(PA) IS CONSISTENT. For anybody to be whining "but it's unsound" is just that -- whining. The IMPORTANT point here is NOT, as Isaacson has alleged, a DIFFERENCE between the way the parallel postulate relates to models of geometry and the way G(PA) relates to models of PA. The IMPORTANT part is the SIMILARITY between the way they relate -- since neither of them is provable from the (other) axioms, BOTH of them are true in some models and false in others. > And on *that* assumption, which I take to be > still in force when we comments on the proof he sketches, In the section I quoted, he iS NOT sketching any proof -- I am quoting from AFTER that. In the section I quoted, he is drawing a (wrong) analogy between Con(PA) and the parallel postulate. > the > canonical Gödel sentence will indeed be true on the standard > interpretation. But proving that is quite beyond the scope of the material. You have to go somewhere like ZF and epsilon_0-induction to prove THAT. > > Hold on, hold on. You've rather ripped this out of context from Dan's > > lecture notes. He two paragraphs earlier introduces the assumption > > that we are dealing with systems of arithmetic S which are *sound* > > (and hence consistent). > > Come ON. That is a lame dodge on YOUR part. Well, I think not. But it isn't my job to defend someone else's informal lecture notes: so while I still think you are misreading Dan, let's agree to differ about the correct reading of his notes. > > the > > canonical Gödel sentence will indeed be true on the standard [quoted text clipped - 3 lines] > You have to go somewhere like ZF and epsilon_0-induction to prove > THAT. But again, what *I* said was that the canonical Gödel sentence *of a sound theory* will be true on the standard interpretation. That follows immediately, without going somewhere like ZF, from 1) For any theory S, the canonical Gödel sentence G_S is true on the standard interpretation if and only if S doesn't prove G_S. [By construction] and 2) For any sound theory S, S doesn't prove G_S [By Gödel's semantic argument for incompleteness] > But again, what *I* said was that the canonical > Gödel sentence *of a sound theory* will be > true on the standard interpretation. This is begging all sorts of questions and mixing all sorts of notions. There is a PRIOR question that you are continuing to lamely dodge and that Isaacson didn't even realize he was confronting: WHAT DOES IT *mean*, in GENERAL, for a "theory" to BE "sound"?? In the case of arithmetic, here, THERE ALREADY IS a standard interpretation. A standard MODEL. ANY theory over the same language as those axioms is sound BY DEFINITION if AND ONLY IF it AGREES with the standard interpretation! THAT IS THE DEFINITION of "sound"! > That follows immediately, OF COURSE it follows immediately that any sentence in any sound theory is true in the standard model, SINCE "sound" is DEFINED as "matching the standard model"! THAT IS *NOT* the point! > 1) For any theory S, the canonical Gödel sentence > G_S is true on the > standard interpretation if and only if S doesn't prove G_S. [By > construction] You have to go to ZF to say ANYthing whatsoEVER about "the standard interpretation". Good grief. What is constructible will occur IN THE NONSTANDARD INTERPRETATIONS AS WELL. > and > > 2) For any sound theory S, > S doesn't prove G_S [By Gödel's semantic > argument for incompleteness] Why do you say "sound" here? For ANY theory rich enough to formulate Con(S) or G_S at all, S doesn't prove G_S. > For ANY theory rich enough to formulate Con(S) > or G_S at all, S doesn't prove G_S. Formal theory. I was thinking about a theory as the closure of a recursive axiom-set. That some random collection of strings that is not even r.e. even qualifies as "a theory" (at all) is an abuse of language. I am every bit as opposed to referring to a NON-formal theory as "a theory" as you are to referring to a formal language as "a language". > Oh? Why does it force us to ask that?? Actually I do know Dan, and of > course we'd both agree with Torkel Franzen's point. Oh, stop lying. The whole reason I quoted it is that Isaacson is directly contradicting TF's point here. > > Oh? Why does it force us to ask that?? Actually I do know Dan, and of > > course we'd both agree with Torkel Franzen's point. > > Oh, stop lying. Oh, come on with the lying already. That really _alienates_ people, which is not what you want. I'm sure you can come up with a different article of obloquy that will be at once sufficiently caustic and cathartic! I have confidence in your resourcefulness. > The whole reason I quoted it is that > Isaacson is directly contradicting TF's point here. -- hz > Oh, come on with the lying already. > That really _alienates_ people, > which is not what you want. You really don't know what I want. What I want, whenever I accuse somebody of having lied, is for him to admit that he has lied. Any other reaction is fundamentally simply another lie. > I'm sure you can come up with a different article of obloquy that > will be at once sufficiently caustic and cathartic! I'm not trying to be caustic. I'm trying to get people to stop lying. > > Oh, come on with the lying already. > > That really _alienates_ people, > > which is not what you want. > > You really don't know what I want. It's my guess that you don't especially want to alienate people. The smart people, anyway. > What I want, whenever I accuse somebody of having lied, > is for him to admit that he has lied. Any other reaction is [quoted text clipped - 5 lines] > I'm not trying to be caustic. > I'm trying to get people to stop lying. I sympathize with your motive. -- hz > > > I'm sure you can come up with a different article of obloquy that > > > will be at once sufficiently caustic and cathartic! I retorted, > > I'm not trying to be caustic. > > I'm trying to get people to stop lying. > > I sympathize with your motive. Well, that will get you this: "The smart people", the ones we need not to alienate, are a little old. They are a little set in their ways. One of the downsides of being smart is that you are better able than the non-smart to tolerate complexity. The downside of that is that when complexity has gotten unreasonable and needs to be pruned, you may not be aware of this -- being smart, you are able to cope with unusuall high levels of complexity without getting upset. The received/conventional wisdom is flawed. This is a truism. EVERYthing is flawed to some degree. MY point is that fixing flaws in "the received/conventional wisdom" has a potentiall broad payoff simply because convention by definition has been broadly adopted. Concomitantly, this sort of change is difficult to effect because conventions are, again by definition, DEEPLY as well as broadly entrenched. The particular argument I was having with P.Smith (I have been having it with everybody else, too, since I arrived, ESPECIALLY TF) was about "sound"ness of models of PA. This is almost equivalent to an argument about the "standard"ness of the standard model. It is also an argument about a standard/default interpretation for a first-order language. There is something to be said for this when you are focusing on the model or the interpretation. But to call a *theory* sound or unsound IS JUST STUPID. PA is sort of special in that the first-order standard/intended/sound/ true interpretation is blessed by the 2nd-order categoricity OF THE SAME language. But that is not the only way to distinguish the interpretation in question, and, far more importantly, if you are not ACTUALL GOING TO GO to 2nd-order, if you are going to continue talking about stronger and stronger 1st-order theories, then what happens at 2nd-order simply is not relevant, and in toto, ALL the special features of the standard model TOGETHER STILL DO NOT ADD up to a REASONABLE reason for deprecating the other models as "unsound". The problem with the (older) smart people is simply that they are *used* to speaking otherwise. > The particular argument I was having with P.Smith (I have been having > it with everybody else, too, since I arrived, ESPECIALLY TF) was about [quoted text clipped - 5 lines] > model or the interpretation. But to call a *theory* sound or unsound IS > JUST STUPID. Yes. There is a deeply-rooted reluctance to let go of the notion of "mathematical truth" as is "standardly" understood. Some special pleading in favor of the PA-ZFC "standard" notions is that they are _sufficient_ to get around in the much larger universe of interpretations that are possible. One size fits all, in a sense. I don't see a willingness, a philosophical willingness, to grasp the situation in its fullest generality -- that all consistent systems are equally "sound". It has been said that since Galileo mankind has been moving from the center of the universe to x. To become at home in the void is the great paradigm shift. -- hz > I don't see a willingness, a philosophical willingness, to grasp > the situation in its fullest generality -- that all consistent > systems are equally "sound". My point is simply that one wonders where that is *coming* from. The traditional bias is in favor of "interpreted languages", which is indefensible BECAUSE the whole import of provability is that interpretations do NOT matter. But I don't think "sound" needs to be discarded ALtogether; my point is that proponents of the notion need to come up with some rational generalizable criterion for soundness. Why does no one talk about "unsound" models of geometry or set theory? As soon as they can come up with examples of *those*, we can rehabilitate the distinction. > It has been said that since Galileo mankind has been moving from > the center of the universe to x. Oops, I meant "Copernicus". -- hz > (It is one anyone > teaching this stuff stresses: sure, we need to assume that the system > we are dealing with is indeed consistent if we are get to see that the > canonical Gödel sentence is true on the standard interpretation.) No, you don't. You DON'T see that the system is consistent because you can't prove it, period. You don't know that the standard interpretation or any other interpretation EVEN EXISTS, because if one did, it would, by existing, prove consistency. Everything you think you want to say about "the standard interpretation" is something you actually wind up proving in ZFC after invoking the (minimal) set satisfying the axiom of infinity. |
Reply to this Message |
 Sign In
 Join
 My Latest Posts My Monitored Threads My Blog My Photo Gallery My Profile My Homepage Start New Thread Enable EMail Alerts Rate this Thread
|
©2010 Advenet LLC Privacy Policy - Terms of Use
This website includes both content owned or controlled by Advenet as well as content owned or controlled by third parties.