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Math Forum / Mathematics / Mathematical Logic / October 2008Yet another disproof of the diagonal argument
This is a disproof of the "diagonal argument" by induction, although I'd rather call it a proof of unsoundness: yet another one, that is. It's actually a shortened version of an argument I have presented in the thread "A consideration concerning the diagonal argument of G. Cantor", that there has gone unnoticed. Here it goes again, in a more synthetic form: Consider the inductive sequence of lists (a formal definition can be found in [1]): n=1 --- 0 1 n=2 --- 00 01 10 11 n=3 --- 000 001 010 011 100 101 110 111 n=4 --- 0000 0001 0010 0011 0100 0101 0110 0111 1000 1001 1010 1011 1100 1101 1110 1111 and so on. Over finite induction we see that every list is simply not square, and that any anti-diagonal at each step, and actually any combination of digits at each step, must be in the list for that step. [] In the limit case, the length of the string may still be incomparable to the length of the list, but this -I suppose- shall be stated in terms of an impossible bijection, because the diagonal argument in itself is (if I am not too mistaken) apparently and provably invalid, the above just being yet another disproof, namely a counter-example given by the simplest inductive construction. Thanks for any (constructive) feedback. -LV [1] See "The Computable Reals (beta)", and the definitions in the code linked here: http://groups.google.co.uk/group/sci.math/msg/a9b219e4e10cc56c?hl=en In article <ccf5c734-cb8d-408d-ad61-10b739ecefef@e53g2000hsa.googlegroups.com>, > This is a disproof of the "diagonal argument" by induction, although > I'd rather call it a proof of unsoundness: yet another one, that is. [quoted text clipped - 49 lines] > > and so on. Each of the above can be extended with an endless string of zeroes, and all resulting lists combined into a single huge list whose Cantor "anti-diagonal" will differ from each and every one of the listed strings. > Over finite induction we see that every list is simply not square, and > that any anti-diagonal at each step, and actually any combination of [quoted text clipped - 4 lines] > terms of an impossible bijection, because the diagonal argument in > itself is (if I am not too mistaken) apparently and provably invalid, You are too mistaken. > the above just being yet another disproof, namely a counter-example > given by the simplest inductive construction. Cantor's "anti-diagonal"proof is still valid for any finite, or even countably infinite, list of lists, since the members of such lists can always be rearranged into a single list before the 'anti-diagonal" is constructed. > Thanks for any (constructive) feedback. The construction of a single list from at most countably many lists, each of at most countably many entries, is well known. At least to mathematicians. Thus no splitting up of strings into separate lists, as is attempted above, is of any consequence. Cantor's theorems still hold. > In article > <ccf5c734-cb8d-408d-ad61-10b739ece...@e53g2000hsa.googlegroups.com>, [quoted text clipped - 57 lines] > "anti-diagonal" will differ from each and every one of the listed > strings. No, it can't. And you might even translate your nonsense to proper mathematical definitions, and still that will have nothing to do with the OP and, a fortiori, won't show any flaw in the given proof. -LV In article <07fc1892-4658-4e1c-abf2-255b81a38b4e@m45g2000hsb.googlegroups.com>, > > In article > > <ccf5c734-cb8d-408d-ad61-10b739ece...@e53g2000hsa.googlegroups.com>, [quoted text clipped - 59 lines] > > No, it can't. Which of the finite strings above is incapable of being extended to form a longer string? That you may not want to let them to be extended does not make them incapable of being extended. And how are your finite strings relevant to Cantors statement which refers only to infinite strings? >And you might even translate your nonsense to proper > mathematical definitions I am not the one who is having trouble with definitions. My usage conforms at least to standard mathematical usage. Yours does not. > and still that will have nothing to do with > the OP and, a fortiori, won't show any flaw in the given proof. There is no flaw in Cantor's given proof. Nor virtue in LV's counter arguments. > This is a disproof of the "diagonal argument" by induction, although > I'd rather call it a proof of unsoundness: yet another one, that is. [quoted text clipped - 62 lines] > > Thanks for any (constructive) feedback. ************************************************************** Correction : the above being yet another amateur, pretty sloppy try by a most probably amateur non-mathematician to find a counter-example to a well-based, old, sound and beautiful mathematical proof: 1) Why do you think you can arrange numbers the way you want AND THEN apply the diagonal argument? 2) What do you mean by "in the limit case"?? 3) What do you mean by "the length of the string may still be incomparable to the length of the list"? 4) What do you mean that 3 "shall be stated in terms of an impossible bijection, because the diagonal argument in itself is (if I am not too mistaken) apparently and provably invalid,..."?? So you're saying that something in your proof cannot be true because what you want to proof is apparently and probably invalid?? This makes no sense, of course... Regards Tonio > > This is a disproof of the "diagonal argument" by induction, although > > I'd rather call it a proof of unsoundness: yet another one, that is. [quoted text clipped - 68 lines] > a most probably amateur non-mathematician to find a counter-example to > a well-based, old, sound and beautiful mathematical proof: Correction? I am rather touched. That proof and argument I genuinely find invalid, and it stays at the foundations of mathematical and logical thinking, in a territory that is not that far from my specific competences. In more specific domains of mathematics and mathematical logic, I am surely at best a beginner. > 1) Why do you think you can arrange numbers the way you want AND THEN > apply the diagonal argument? The diagonal argument is simply not stringent as to the definition of the list. I provide a counter-example. Would you say the case I have provided is invalid? If so, why? > 2) What do you mean by "in the limit case"?? The proof ends at the "[]". This already belongs to the ideas and questions. The limit case here is of course the limit of the inductive process. > 3) What do you mean by "the length of the string may still be > incomparable to the length of the list"? Ideas and questions. > 4) What do you mean that 3 "shall be stated in terms of an impossible > bijection, because the diagonal argument in itself is (if I am not too > mistaken) apparently and provably invalid,..."?? Ideas and questions. I mean that invalidating the diagonal argument is not enough to state the countability of the reals, but all definitions should be really checked and restated, so again: just ideas and questions. > So you're saying that something in your proof cannot be true because > what you want to proof is apparently and probably invalid?? This > makes no sense, of course... The proof ends at the "[]". Of course I am an amateur, if that means passionate, serious and sincere. -LV ......................................................... > > Correction : the above being yet another amateur, pretty sloppy try by > > a most probably amateur non-mathematician to find a counter-example to [quoted text clipped - 5 lines] > competences. In more specific domains of mathematics and mathematical > logic, I am surely at best a beginner. ************************************************************ Great, so: why do you find that argument and proof (I supose Cantor's Theorem and its diagonal proof, by Cantor himself, too) "invalid"? Where do you find a possible flaw or invalid statement in Cantor's Diagonal Proof? ************************************************************ > > 1) Why do you think you can arrange numbers the way you want AND THEN > > apply the diagonal argument? > > The diagonal argument is simply not stringent as to the definition of > the list. I provide a counter-example. Would you say the case I have > provided is invalid? If so, why? ******************************************************************* I supose that by "astringent" you mean that the diagonal argument doesn't require the supposed list of reals to be be a special, specific and particular one, and that's right. But then why didn't you write every number in your lists with zeroes at the beginning of it? For example, in list 1 we get 0 = ...000 and 1 = ...001 , etc. That way you have no problem with the squaring stuff, which seems to be a central point in your argument... ******************************************************************* > > 2) What do you mean by "in the limit case"?? > > The proof ends at the "[]". This already belongs to the ideas and > questions. The limit case here is of course the limit of the inductive > process. *********************************************************** Ok, let's try to be a little more accurate here: the not-so-clear argumentation that you call proof ends in [], according to you, and it is not clear to me what and how do you end, and why at the end of it you think you have shown that the diagonal argument in invalid. And again: what is the (or a) limit of some inductive process?? ********************************************************** > > 3) What do you mean by "the length of the string may still be > > incomparable to the length of the list"? > > Ideas and questions. *************************************************************** Why do you think that's an answer to my question? ************************************************************** > > 4) What do you mean that 3 "shall be stated in terms of an impossible > > bijection, because the diagonal argument in itself is (if I am not too [quoted text clipped - 13 lines] > Of course I am an amateur, if that means passionate, serious and > sincere. ****************************************************************** No. In this case, amateur means someone that hasn't studied mathematics (as something like 99-100% of the anticantorian cranks in this forum, without calling you by this a crank) and thus cannot construct a supposed MATHEMATICAL proof, or disproof, of a MATHEMATICAL statement. Passion, seriousness and sincerity aren't good enough to come in place of mathematical soundness and accuracy, though they are very nice personal characteristics. Regards Tonio LudovicoVan wrote: >> Of course I am an amateur, if that means passionate, serious and sincere. > No. In this case, amateur means someone that hasn't studied > mathematics (as something like 99-100% of the anticantorian cranks in > this forum, without calling you by this a crank) and thus cannot > construct a supposed MATHEMATICAL proof, or disproof, of a > MATHEMATICAL statement. I'm glad you clarified your position. I consider myself an amateur, but not of the variety you describe. I have studied math (I have a B.S. in Mathematics, in fact), and continue to dabble in it, from the "outside", as it were. On the other hand, I'm not a professional mathematician, i.e., I do not get paid to produce theorems, research, or teach in math. (Unless you count software engineering as a form of professional math, which I don't.) So based on your definition, I am of the variety of mathematician who falls somewhere between your "amateur" and "professional". How about "Humble dabbler in the mathematical arts"? > Passion, seriousness and sincerity aren't good enough to come in place > of mathematical soundness and accuracy, though they are very nice > personal characteristics. Indeed. I am reminded of a "Peanuts" cartoon in which Linus asks, "But how can we lose when we're so sincere?" >> Of course I am an amateur, if that means passionate, serious and >> sincere. [quoted text clipped - 4 lines] > construct a supposed MATHEMATICAL proof, or disproof, of a > MATHEMATICAL statement. http://www.thefreedictionary.com/amateur: 1. A person who engages in an art, science, study, or athletic activity as a pastime rather than as a profession. 2. [ ... ] 3. One lacking the skill of a professional, as in an art. There are two definitions of amateur in play. The trend has increasingly been to take it to mean the latter definition in the majority of cases, indicating the intent of the former definition via terms like "recreational." Probably a better common definition is this: "One who lacks formal instruction in a skill." "Amateur" has generally come to have a negative or at least a condescending tone in that the addressee usually: a. is trying to overturn well-established norms (like Cantor's diagonal argument, Gödel's Incompleteness Theorem), b. has little to no formal experience or instruction in the subject, and c. accompanies said claim with such profusions of grandeur as "I am the FIRST person to COMPLETELY understand this." > >> Of course I am an amateur, if that means passionate, serious and > >> sincere. [quoted text clipped - 24 lines] > c. accompanies said claim with such profusions of grandeur as "I am the > FIRST person to COMPLETELY understand this." Origin: 1775-85; < F, MF < L amator lover, equiv. to ama- (s. of amare to love) + -tor An amateur acts from love, not for pay. -- hz > .........................................................> > Correction : the above being yet another amateur, pretty sloppy try by > > > a most probably amateur non-mathematician to find a counter-example to [quoted text clipped - 12 lines] > Where do you find a possible flaw or invalid statement in Cantor's > Diagonal Proof? I do have some prelimiary ideas on this, but I won't be able to talk about it until the OP has been settled down. I need this result to proceed: the diagonal argument underlies some of the most fundemental arguments in mathematics and logic, Cantor's theorem of course being one of them. > ************************************************************ > [quoted text clipped - 14 lines] > That way you have no problem with the squaring stuff, which seems to > be a central point in your argument... "Astringent"? Anyway, you too, please note that you can't just say "square it" or "add leading zero's" or anything: you have to provide different definitions. And then it is not anymore the OP that you are talking about. > ******************************************************************* > [quoted text clipped - 11 lines] > you think you have shown that the diagonal argument in invalid. > And again: what is the (or a) limit of some inductive process?? You may be right: I'll try to be a little bit more explicit in another post. In any case, your question on what the limit of an inductive process is still makes no sense out of the context of the comments that *follow* the proof. There is simply no mention and no need to mention limits in the proof. > ********************************************************** > [quoted text clipped - 6 lines] > > Why do you think that's an answer to my question? You are asking me to be rigurous on what was not meant be. You surely understand the difference between a statement of a proof and a statement belonging to some open comments and ideas. And that is why, BTW, I am so reticent in discussing anything beyond the OP: we get lost in confusion too easily in this group. > ************************************************************** > [quoted text clipped - 20 lines] > No. In this case, amateur means someone that hasn't studied > mathematics And when exactly would you declare that one "has studied" mathematics? I prefer to assume that we can be "professionals" since day one, as a way and a method, despite that we are always beginners when we approach something new. > (as something like 99-100% of the anticantorian cranks in > this forum, without calling you by this a crank) and thus cannot [quoted text clipped - 3 lines] > of mathematical soundness and accuracy, though they are very nice > personal characteristics. There surely is the proverbial "learning curve". -LV In article <49121f76-ccf7-4e71-9aaa-2b6be679d2ee@m36g2000hse.googlegroups.com>, > > > Of course I am an amateur, if that means passionate, serious and > > > sincere. [quoted text clipped - 8 lines] > way and a method, despite that we are always beginners when we > approach something new. Passion, seriousness and sincerity only count in mathematics when accompanied by precision, logic and clarity, all of which you lack. ******************************************************************* ....................................................... > > > > 2) What do you mean by "in the limit case"?? > [quoted text clipped - 13 lines] > > In any case, your question on what the limit of an inductive process is still makes no sense out of the context of the comments that *follow* the proof. There is simply no mention and no need to mention limits in the proof. ****************************************************************** ****************************************************************** The "comments" following what you call "proof" seemed to be a rather important part of the same, and that's why, perhaps, many didn't quite understand that what you call proof ended in []. And YOU were the one "commenting" about "limit cases" and then again, answering one of my questiosn, you wrote "The limit case here is of course the limit of the inductive process." which, mathematically, means the hell knows what, and that's the reason I asked you. Finally, I think it makes a lot of sense to ask about anything someone wrote in the proof, or disproof, of something or in the sequel of it. ****************************************************************** ****************************************************************** > > ********************************************************** > [quoted text clipped - 9 lines] > You are asking me to be rigurous on what was not meant be. You surely understand the difference between a statement of a proof and a > statement belonging to some open comments and ideas. And that is why, BTW, I am so reticent in discussing anything beyond the OP: we get lost in confusion too easily in this group. **************************************************************** **************************************************************** Oh, so your OP wasn't meant to be a proof but just a "statement belonging to some open comment and ideas"? So you didn't mean to be rigurous in your purposed "disproof" of Cantor's Theorem's diagonal proof? Then your OP's bombastic title is exaggerated, don't you think? And of course, this that you wrote in your OP: "because the diagonal argument in itself is (if I am not too mistaken) apparently and provably invalid, the above just being yet another disproof, namely a counter-example given by the simplest inductive construction." seems pretty affirmative to me, and not only "a comment", not to mention that the word "another" above points towards the existence of, at least, another such disproof, about which I'll dare to ask: which one, please? ************************************************************** ************************************************************** ...................................... > > No. In this case, amateur means someone that hasn't studied > > mathematics > > And when exactly would you declare that one "has studied" mathematics? > I prefer to assume that we can be "professionals" since day one, as a way and a method, despite that we are always beginners when we > approach something new. *********************************************************** *********************************************************** I'd declare that someone has studied mathematics when, and please allow me to indulge myself in this since I'm not going to be very subtle, that someone ACTUALLY studied mathematics. Now, perhaps your question could better be expressed as "how can you say someone has studied, or has not studied, mathematics?", which was in fact what prompted me to write that thing about amateurs, but then I'll write that it is obvious out of some purposed "proofs" and "disproofs" from people who haven't studied mathematics, since the usually have not the slightest and palest idea what a mathematical proof looks like, leave alone how to construct and end one. And please note I wrote "study mathematics", not going through graduate school, or even doing a major, in mathematics. Regards Tonio (as something like 99-100% of the anticantorian cranks in > > this forum, without calling you by this a crank) and thus cannot > > construct a supposed MATHEMATICAL proof, or disproof, of a [quoted text clipped - 4 lines] > > There surely is the proverbial "learning curve". > And when exactly would you declare that one "has studied" mathematics? When he knows propositional and predicate calculus, that's when. > I prefer to assume that we can be "professionals" since day one, as a > way and a method, despite that we are always beginners when we > approach something new. You, unfortunately, did not have sense enough to approach this that way. When you know you are new, you ask questions, because you know you need to learn. You, unfortunately,do not know this. You do not know that you need to be ASKing me what a proof is instead of TELLing me. You do not know that you need to be ASKing me what an inductive definition is instead of TELLing me. > > (as something like 99-100% of the anticantorian cranks in > > this forum, without calling you by this a crank) and thus cannot [quoted text clipped - 5 lines] > > There surely is the proverbial "learning curve". But that does not become relevant until you ask someone to teach you, instead of ridiculously claiming to already know. trimpi In article <aeb2d020-82c0-438c-80c9-c917d8057be3@m45g2000hsb.googlegroups.com>, > Of course I am an amateur, if that means passionate, serious and > sincere. It certainly does not mean that you are correct. There are all sorts of people who are "passionate, serious and sincere" and WRONG! You appear to be one of them. > This is a disproof of the "diagonal argument" by induction, although > I'd rather call it a proof of unsoundness: yet another one, that is. [quoted text clipped - 68 lines] > linked here: > http://groups.google.co.uk/group/sci.math/msg/a9b219e4e10cc56c?hl=en What position is 1/9 = 0.1111.... in the list? If you have a list of all Reals, then every Real has to appear somewhere - at some position n. What n gives you 1/9 ? On 28 Sep, 11:44, "Peter Webb" <webbfam...@DIESPAMDIEoptusnet.com.au> wrote: > > This is a disproof of the "diagonal argument" by induction, although > > I'd rather call it a proof of unsoundness: yet another one, that is. [quoted text clipped - 75 lines] > > What n gives you 1/9 ? Simply no mention of Reals in the OP, we talk about the diagonal argument here. You might rather be interested in the other thread on the infinite binary tree. -LV In article <7758aaa7-f7b7-4ebd-bc1c-320147cee2b7@y38g2000hsy.googlegroups.com>, > > What position is 1/9 = 0.1111.... in the list? > > [quoted text clipped - 8 lines] > You might rather be interested in the other thread on the infinite > binary tree. Since the "Subject" is whether the unmentioned reals can be listed, as Cantor's diagonal argument claims, the reals are in it whether you mention them or not. And that "binary tree" argument is equally fallacious. You've basically shown that a diagonal argument can't be used to show that the non-negative integers are uncountable--which is not at all surprising, since they are countable. You haven't shown anything at all about the applicability of diagonal arguments to other sets.  Signature--Tim Smith >This is a disproof of the "diagonal argument" by induction, although >I'd rather call it a proof of unsoundness: yet another one, that is. [quoted text clipped - 53 lines] >that any anti-diagonal at each step, and actually any combination of >digits at each step, must be in the list for that step. [] True, and irrelevant. >In the limit case, the length of the string may still be incomparable >to the length of the list, but this -I suppose- shall be stated in [quoted text clipped - 4 lines] > >Thanks for any (constructive) feedback. Nothing above has anything to do with the diagonal argument. There is no "limit case" there. The set of all binary strings of infinite length simply is _not_ the limit as n -> infinity of the set of binary strings of length n. >-LV > >[1] See "The Computable Reals (beta)", and the definitions in the code >linked here: >http://groups.google.co.uk/group/sci.math/msg/a9b219e4e10cc56c?hl=en David C. Ullrich "Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to." (John Jones, "My talk about Godel to the post-grads." in sci.logic.) > >This is a disproof of the "diagonal argument" by induction, although > >I'd rather call it a proof of unsoundness: yet another one, that is. [quoted text clipped - 70 lines] > The set of all binary strings of infinite length simply is _not_ > the limit as n -> infinity of the set of binary strings of length n. Oh, come on Dr Ullrich, you have been brainwashed. Of course this may not hold under conventional _finite_ induction, but unleashing the full power of Transfinite Induction (technically not quite what you think TI is...) enables, enables, well, you know, anything. Just like interplanetary travel -- the sky's the limit. Sig. Ludovico must be congratulated on a brilliant success. Unlike many cranks who drone on for page after page, he can distill his confusion into a single paragraph. Such verbal economy! Brian Chandler > On Sat, 27 Sep 2008 23:34:10 -0700 (PDT), LudovicoVan > [quoted text clipped - 70 lines] > Nothing above has anything to do with the diagonal argument. > There is no "limit case" there. You have missed that the proof ends at the "[]". -LV In article <529e955d-6b7c-460b-a84a-dcbb10ab8dd4@s50g2000hsb.googlegroups.com>, > > <ju...@diegidio.name> wrote: > > >This is a disproof of the "diagonal argument" by induction, although [quoted text clipped - 70 lines] > > You have missed that the proof ends at the "[]". Actually, by the "[]", there is no proof as yet completed. Nor, as far as I can see, even started. > -LV >> <ju...@diegidio.name> wrote: >> >This is a disproof of the "diagonal argument" by induction, although [quoted text clipped - 70 lines] > >You have missed that the proof ends at the "[]". Uh, right. If the proof ends at the "[]" then it has even less to do with the standard diagonal argument. At the [] I said "True, and irrelevant", which is exactly the status at that point. Exactly what is it you imagine you've proved here? >-LV David C. Ullrich "Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to." (John Jones, "My talk about Godel to the post-grads." in sci.logic.) > The set of all binary strings of infinite length simply is _not_ > the limit as n -> infinity of the set of binary strings of length n. Well, it -is- the (contravariant) limit (in the category of sets) of the sequence 2^1 <- 2^2 <- ... where 2^n denotes the set of binary strings of length n, and the arrows are the canonical surjections ('restriction to initial segments', so to say). SignatureCheers, Herman Jurjus > > The set of all binary strings of infinite length simply is _not_ > > the limit as n -> infinity of the set of binary strings of length n. [quoted text clipped - 3 lines] > of length n, and the arrows are the canonical surjections ('restriction > to initial segments', so to say). ****************************************************************** Yes, wonderful...and yet it still is NOT the limit as n --> oo of the set of binary strings of legth n, just as David wrote. Regards Tonio Ps. "Contravariant limit"....really! The OP is by an amateur: what do you think he thinks about functor, contraviarant stuff and category theory? >>> The set of all binary strings of infinite length simply is _not_ >>> the limit as n -> infinity of the set of binary strings of length n. [quoted text clipped - 14 lines] > you think he thinks about functor, contraviarant stuff and category > theory? All the more reason to inform him about this. Lest he thinks that mathematics is still in the stage it was in in 1874. It's a good thing to tell newbies and crackpots (if possible) that their intuitions are not only sensible, but also formalizable with ordinary, standard mathematical means. At least that's better than to give them the (erroneous) impression that mathematicians are a bunch of narrowminded twats with restricted imaginatory capabilities. SignatureCheers, Herman Jurjus > >>> The set of all binary strings of infinite length simply is _not_ > >>> the limit as n -> infinity of the set of binary strings of length n. [quoted text clipped - 23 lines] > the (erroneous) impression that mathematicians are a bunch of > narrowminded twats with restricted imaginatory capabilities. ****************************************************** When we're talking about really huge crackpots , I wouldn't give much of a damn what they think about mathematics and/or mathematicians: if they step in this forum, make bombastic, completely baseless and utter stupid announcements like "Cantor's Theorem debunked", "FLT's proof with mathematics of an 8 y.o. kids", "There are no infinite sets", etc., then they mess up with things waaaaaay above their head, and....well, it's fun to try to correct them. After 2-3 attempts are made trying to open some crank's eyes about why what he thinks is true is NOT true from a mathematical point of view, I don't really care much about what he thinks or not: it's sheer joy, sometimes, to deal with her/him; of course, other times it gets boring. Fun, that's all...and ocassionally, when dealing with crackpots posts, perhaps learning a new worthy thing from other poster, which usually is not the case when dealing with cranks. BTW, and for the record, I do NOT consider Ludovico neither a crackpot nor a crank, though sometimes in this thread he has gotten pretty close to the trolling limits. I think he's just a nice amateur guy trying to make a point about his thoughts, but he lacks basic training in maths, that's all. Too bad he chose bombastic, nonsensical words fo this. Regards Tonio Tonico a écrit : >>>>> The set of all binary strings of infinite length simply is _not_ >>>>> the limit as n -> infinity of the set of binary strings of length n. [quoted text clipped - 42 lines] > nor a crank, though sometimes in this thread he has gotten pretty > close to the trolling limits. Like when he suddenly take a typical anti-psychiatrist extreme posture? How much o you need to realize you *have* been trolled? I think he's just a nice amateur guy > trying to make a point about his thoughts, but he lacks basic training > in maths, that's all. Too bad he chose bombastic, nonsensical words fo > this. > > Regards > Tonio > >>> The set of all binary strings of infinite length simply is _not_ > >>> the limit as n -> infinity of the set of binary strings of length n. [quoted text clipped - 17 lines] > All the more reason to inform him about this. > Lest he thinks that mathematics is still in the stage it was in in 1874. That stage is a golden age compared to the current one. > It's a good thing to tell newbies and crackpots (if possible) that their > intuitions are not only sensible, but also formalizable with ordinary, > standard mathematical means. The real crackpots are these self claiming mathematicians who are incapable to contribute anything but their consistent sh.tting. > At least that's better than to give them > the (erroneous) impression that mathematicians are a bunch of > narrowminded twats with restricted imaginatory capabilities. No, and it's not just an "impression", it's as objective as that the water is wet: these are just a bunch of idiots, nothing to do with true mathematics and true accademia. -LV In article <9aaf981f-a77b-4aba-952b-7a26378670ba@z66g2000hsc.googlegroups.com>, > > >>> The set of all binary strings of infinite length simply is _not_ > > >>> the limit as n -> infinity of the set of binary strings of length n. [quoted text clipped - 19 lines] > > That stage is a golden age compared to the current one. It was no more a mathematical golden age than it was a medical golden age. > > It's a good thing to tell newbies and crackpots (if possible) that their > > intuitions are not only sensible, but also formalizable with ordinary, > > standard mathematical means. > > The real crackpots are these self claiming mathematicians who are > incapable to contribute anything but their consistent sh.tting. to everyone has the capacity to grasp mathematics. But it only "tastes" bad to those who are incapable of grasping it. > > At least that's better than to give them > > the (erroneous) impression that mathematicians are a bunch of [quoted text clipped - 3 lines] > water is wet: these are just a bunch of idiots, nothing to do with > true mathematics and true accademia. And just what are julio's supposed credentials as an expert on what constitutes either true mathematics or true academia? So far, nothing he has posted supports him having any expertise in either area. > > >>> The set of all binary strings of infinite length simply is _not_ > > >>> the limit as n -> infinity of the set of binary strings of length n. [quoted text clipped - 34 lines] > water is wet: these are just a bunch of idiots, nothing to do with > true mathematics and true accademia. May you enlighten us as to the reasons, then, which prompt you to insist in communicating with them, being, as it seems, that so far on sci.math you have yet to deal with but idiots? One would imagine that, given the apparent familiarity you have with true mathematics and true academia, you would prefer that to this, and spend your time rather there than here, with much loftier company. -- m On 30 Sep, 03:53, Mariano Suárez-Alvarez <mariano.suarezalva...@gmail.com> wrote: > > > >>> The set of all binary strings of infinite length simply is _not_ > > > >>> the limit as n -> infinity of the set of binary strings of length n. [quoted text clipped - 43 lines] > you would prefer that to this, and spend your time > rather there than here, with much loftier company. If you and all the morons just made me the favour to f.ck off from this and all other threads I am interested in, that would be simply perfect. On the threads of mine, I even think I have some legitimate right at it. BTW, have you read my reply to Mr Nakhash? Never mind, just a rethorical question. -LV In article <946ab190-3fe3-40d0-b77f-2ebdad6c29eb@x41g2000hsb.googlegroups.com>, > If you and all the morons just made me the favour to f.ck off from > this and all other threads I am interested in, that would be simply > perfect. On the threads of mine, I even think I have some legitimate > right at it. I, for one, have been posting here on sci.math for about 10 years, so if anyone is to "f.ck off" from all the threads in sci.math I am interested in, it should be you. > >>> The set of all binary strings of infinite length simply is _not_ > >>> the limit as n -> infinity of the set of binary strings of length n. [quoted text clipped - 23 lines] > the (erroneous) impression that mathematicians are a bunch of > narrowminded twats with restricted imaginatory capabilities. It is probably helpful to direct him toward coalgebras and coninductive definitions of infinite binary strings then. It might actually help him see where he has gone so woefully wrong if he can actually understand it. On 30 Sep, 18:47, leland.mcin...@gmail.com wrote: > > >>> The set of all binary strings of infinite length simply is _not_ > > >>> the limit as n -> infinity of the set of binary strings of length n. [quoted text clipped - 28 lines] > actually help him see where he has gone so woefully wrong if he can > actually understand it. Right, if not in the substance, at least in the approach. It's a confort there still is some (semi-) sanity. -LV In article <4cb2ea01-9c4f-4f7d-ad73-7197a7764555@59g2000hsb.googlegroups.com>, > On 30 Sep, 18:47, leland.mcin...@gmail.com wrote: > > > >>> The set of all binary strings of infinite length simply is _not_ [quoted text clipped - 35 lines] > > It's a confort there still is some (semi-) sanity. But julio, AKA LV, does not have access to any of it. >> The set of all binary strings of infinite length simply is _not_ >> the limit as n -> infinity of the set of binary strings of length n. [quoted text clipped - 3 lines] >of length n, and the arrows are the canonical surjections ('restriction >to initial segments', so to say). How does that have any relevance? David C. Ullrich "Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to." (John Jones, "My talk about Godel to the post-grads." in sci.logic.) >>> The set of all binary strings of infinite length simply is _not_ >>> the limit as n -> infinity of the set of binary strings of length n. [quoted text clipped - 4 lines] > > How does that have any relevance? It refutes your statement. (To put it more bluntly: let's not bash crackpots for the wrong reasons or with the wrong arguments. It's bad pr for mathematics, and it gives crackpots the idea that they've something valuable to contribute.) SignatureCheers, Herman Jurjus Herman Jurjus a écrit : >>>> The set of all binary strings of infinite length simply is _not_ >>>> the limit as n -> infinity of the set of binary strings of length n. [quoted text clipped - 6 lines] > > It refutes your statement. Not really. No more than " "sqrt(+OO) is the limit of the sequence (+oo,+oo/2,+oo/4,...)" is false" is refuted by "it is true in the surreals, with oo denoting w and limit being undesrtood as simplest lower bound" > (To put it more bluntly: let's not bash crackpots for the wrong reasons > or with the wrong arguments. It's bad pr for mathematics, and it gives > crackpots the idea that they've something valuable to contribute.) But this particuliar crackpot is in fact a troll, and best policy here is not to bash with arguments (right or wrong), but with silence :-) > (To put it more bluntly: let's not bash crackpots for the wrong reasons > or with the wrong arguments. It's bad pr for mathematics, and it gives > crackpots the idea that they've something valuable to contribute.) It's not about having something of value to contribute, you terminal idiots: I am here to just "learn and produce". You guys are lost in a demented dream. -LV In article <430f0174-b97f-463d-af10-9cd640daba8e@25g2000hsx.googlegroups.com>, > > (To put it more bluntly: let's not bash crackpots for the wrong reasons > > or with the wrong arguments. It's bad pr for mathematics, and it gives [quoted text clipped - 3 lines] > idiots: I am here to just "learn and produce". You guys are lost in a > demented dream. So far you seem to have learnt nothing and certainly by any mathematical standard you have as yet produced nothing. So that our littlest "demented dreams" have produced more of value that all your learning and producing so far. If you ever intend to either learn or produce here, now would be a good time to start. >>>> The set of all binary strings of infinite length simply is _not_ >>>> the limit as n -> infinity of the set of binary strings of length n. [quoted text clipped - 6 lines] > >It refutes your statement. No it doesn't. The word "limit" is used in various ways - in the sense in which I meant the term the set of infinite binary strings is not the limit of the set of strings of length n. >(To put it more bluntly: let's not bash crackpots for the wrong reasons >or with the wrong arguments. Huh? The fact that this set is not the limit of those sets, in the relevant sense, is _exactly_ why various supposed enumerations of the reals are not in fact enumerations of the reals. >It's bad pr for mathematics, and it gives >crackpots the idea that they've something valuable to contribute.) David C. Ullrich "Understanding Godel isn't about following his formal proof. That would make a mockery of everything Godel was up to." (John Jones, "My talk about Godel to the post-grads." in sci.logic.) > > The set of all binary strings of infinite length simply is _not_ > > the limit as n -> infinity of the set of binary strings of length n. [quoted text clipped - 3 lines] > of length n, and the arrows are the canonical surjections ('restriction > to initial segments', so to say). The should the whole thing, from Cantor's diagonal theorem onwards, be done in topoi terms? > This is a disproof of the "diagonal argument" by induction, although > I'd rather call it a proof of unsoundness: yet another one, that is. [quoted text clipped - 18 lines] > that any anti-diagonal at each step, and actually any combination of > digits at each step, must be in the list for that step. But your lists do /not/ contain their own anti-diagonal elements. For the n=2 list, for example, the list contains 4 elements, therefore the anti-diagonal must contain at least 4 digits. This is obvious, because the anti-diagonal must contain an anti-digit taken from each of the 4 rows of the list. Yet all I see in that list are 2-digit elements. As someone else mentioned, you must consider each element as having a trailing suffix of zero digits in order to properly construct the anti-diagonal from the list. In general, each list n is composed of n-digit elements, for a total of 2^n elements. The anti-diagonal of list n must therefore have 2^n digits, being composed of one anti-digit from each of the 2^n elements of the list. Which means that, by your own definition, your lists cannot possibly contain their own anti-diagonal elements. > In the limit case, ... I don't follow what you mean by the "limit case". Could you provide a definition? > But your lists do /not/ contain their own anti-diagonal elements. No, stop; you're being completely uncharitable. These lists ARE NOT SQUARE, and he CONCEDES this. So YOU Have to concede that SOME sort of TWEAKING of the MEANING of "diagonal" MUST be relevant. The question then becomes whether the particular tweaking herein-invoked was (vs. wasn't) REASONABLE. > For the n=2 list, for example, the list contains 4 elements, > therefore the anti-diagonal must contain at least 4 digits. You are SIMPLY LYING. The list is 4x2. You cannot privilege the length over the width. You allege that the "diagonal" of a 4x2 list must be of width 4. HE IS ALLEGING THAT IT MUST BE OF WIDTH 2. His rationale IS EXACTLY THE SAME AS yours! Of the two dimensions, HE JUST PICKED ONE! > This is obvious, because the anti-diagonal must contain > an anti-digit taken from each of the 4 rows of the list. It is therefore equally obvious that the anti-diagonal must contain an anti-digit taken from EACH OF THE TWO COLUMNS of the list. You MAY NOT do this. You MAY NOT just randomly decide that the rows are more important than the columns, not without LOSING the argument by being FORCED to concede that just randomly deciding that THE COLUMNS ARE MORE IMPORTANT THAN THE ROWS would be equally legitimate (or equally arbitrary, and therefore equally ILlegitimate). In article <b68da9c4-cae8-457e-b171-8b6fb473bcb4@z72g2000hsb.googlegroups.com>, > > But your lists do /not/ contain their own anti-diagonal elements. > [quoted text clipped - 14 lines] > His rationale IS EXACTLY THE SAME AS yours! > Of the two dimensions, HE JUST PICKED ONE! If we are speaking of the "anti-diagonal" of Cantor, that requires that for any n in the list, the nth member have at least n characcters. If those digits are no otherwise supplied, then they may be assumed to be "0" or " ", or whatever you are using as your null character. Thus the list 00 01 10 11 must actually be either based on two characters, "0" and "1" giving "0000" "0100" "1000" "1100" from which 2^4 - 4 = 12 possible strings are missing or three characters, "0", "1" and " ", giving "00 " "01 " "10 " "11 " from which 3^4 - 4 = 77 possible strings are missing. in either case allowing for at least one string missing. > > This is obvious, because the anti-diagonal must contain > > an anti-digit taken from each of the 4 rows of the list. > > It is therefore equally obvious that the anti-diagonal must contain > an anti-digit taken from EACH OF THE TWO COLUMNS > of the list. That is not the way Cantor set it up. His original diagonal argument concerned listing infinite binary strings of two different characters, specifically mappings from N to {m, w}. > You MAY NOT do this. Cantor did it, so you may not refuse to do it when discussing what Cantor did. > You MAY NOT just randomly decide that the rows are more important > than the columns, not without LOSING the argument by being FORCED > to concede that just randomly deciding that THE COLUMNS ARE MORE > IMPORTANT THAN THE ROWS would be equally legitimate (or equally > arbitrary, and therefore equally ILlegitimate). On can always embed any set of finite strings in a set of infinite strings, each extension having the same finite initial string as the one it extends. If this is done so all have the same extention, then the set before and set after naturally biject. Thus one may assume a priori assuming all strings are infinite without loss of generality. > In article > <b68da9c4-cae8-457e-b171-8b6fb473b...@z72g2000hsb.googlegroups.com>, [quoted text clipped - 19 lines] > If we are speaking of the "anti-diagonal" of Cantor, that requires that > for any n in the list, the nth member have at least n characcters. If you can come up with an inductive definition that provides a square list, that might be interesting. Still all this is marginal to the OP. -LV > If you can come up with an inductive definition that provides a square > list, that might be interesting. > > Still all this is marginal to the OP. Induction cannot be used to prove anything about infinity. Here's why: This is the argument of induction. Assume that some fact holds true for a natural (or integer, sometimes) number n. Show that, under this assumption, it will hold true for n+1. Now show that it holds true for a certain x, say x = 0 or x = 1. The fact will therefore hold true for all natural numbers n >= x. Equivalently, the key step is that, the proof that it works for the number n relies on it working for n - 1. Infinity is not a natural number. Infinity minus one is... infinity. Infinity minus any natural number is infinity. If we try to use this inductive step, we will need to show that, to prove it works for infinity, it works for infinity - 1 = infinity. Your argument then reduces to "Assume it works for infinity. [ ... ] It therefore works for infinity," or (in simpler terms), p -> p. It tells us nothing about its actual truth, since we were forced to assume that it was true in the first place. I've mentioned twice in this post, and will mention again, an example of a property that holds true for all natural numbers but not for infinity: that x - 1 =/= x. Basic, intuitive intuitive rules break down at infinity. In short: just because something holds true for all finite numbers does not mean it holds true for infinity. > > If you can come up with an inductive definition that provides a square > > list, that might be interesting. [quoted text clipped - 26 lines] > In short: just because something holds true for all finite numbers does > not mean it holds true for infinity. And how is all this related to the OP? -LV >> In short: just because something holds true for all finite numbers does >> not mean it holds true for infinity. > > And how is all this related to the OP? You're trying to show that, by induction, the list of all real numbers is countable. The definition of an irrational number (the set of which is a subset of real numbers) is that it has a non-repeating, non-terminating decimal representation. Its decimal representation is therefore of infinite length. I'm saying that inducting on the length of decimal representations won't hold for all real numbers and that therefore your proof fails to hold--in other words, attacking the core of your proof. > >> In short: just because something holds true for all finite numbers does > >> not mean it holds true for infinity. [quoted text clipped - 3 lines] > You're trying to show that, by induction, the list of all real numbers > is countable. Again: that is not the list of the "real numbers", surely not in any sense that is essential to a diagonal argument. The diagonal argument works over lists of strings on some alphabet, and in the OP you find the definiton of an inductive list, such that all combinations of digits are present, that is a counter-example to the diagonal argument because all combinations of digits are indeed there, over finite induction: enough for a counter-example to the diagonal argument that itself requires nothing more than the inductive naturals to be stated. Objections are welcome, but yours are missing the point. -LV In article <d7b64331-6cbb-473d-8463-f2dbe0f7b18f@r66g2000hsg.googlegroups.com>, > > >> In short: just because something holds true for all finite numbers does > > >> not mean it holds true for infinity. [quoted text clipped - 6 lines] > Again: that is not the list of the "real numbers", surely not in any > sense that is essential to a diagonal argument. The original diagonal proof was not about numbers at all , but about endless binary strings, the set of which it proves unquestionalbly cannot be counted. It was only later adapted to decimal expansions of real numbers between 0 and 1. > The diagonal argument works over lists of strings on some alphabet, > and in the OP you find the definiton of an inductive list, such that [quoted text clipped - 3 lines] > diagonal argument that itself requires nothing more than the inductive > naturals to be stated. WRONG on two counts. First: the original proof only applies to endless strings, so cannot be refuted by arguments about finite strings. Second: Your notion of induction can, at most, prove things for all finite strings, but never for any infinite strings. > Objections are welcome, but yours are missing the point. If so, then not by nearly as much as you are missing a huge multitude of even more relevant points. In article <d3d95303-48d6-4d36-bd8b-4e1a797e1baa@q9g2000hsb.googlegroups.com>, > > In short: just because something holds true for all finite numbers does > > not mean it holds true for infinity. > > And how is all this related to the OP? Cantor's diagonal proof is stated in terms of and infinite accumulation of infinite binary strings, so infiniteness is inherent in the OP issue, as well as in the stated subject: Yet another disproof of the diagonal argument. In article <db3f53df-2daa-4303-89a9-ad35f898a10b@m36g2000hse.googlegroups.com>, > > In article > > <b68da9c4-cae8-457e-b171-8b6fb473b...@z72g2000hsb.googlegroups.com>, [quoted text clipped - 22 lines] > If you can come up with an inductive definition that provides a square > list, that might be interesting. If you mean a list in which the number of characters in a listed string bijects with the number of strings listed, that doesn't even need induction. Let the function (string) listed in row n be defined by f_n:N -> {0,1}: f_n(m) = 1 for m <= n and f_n(m) = 0 for m > n. > Still all this is marginal to the OP. On the contrary, the original OP was about Cantor's diagonal proof, and so is this one. >> In article >> <b68da9c4-cae8-457e-b171-8b6fb473b...@z72g2000hsb.googlegroups.com>, [quoted text clipped - 23 lines] >If you can come up with an inductive definition that provides a square >list, that might be interesting. Cantor's argument does not require a square list, so it's not clear why our failure to do so it meaningful. The statements you make in your argument are true, but they don't apply to Cantor's proof. That's the fundamental flaw in your argument. Alan SignatureDefendit numerus > Cantor's argument does not require a square list, so it's not clear why > our failure to do so it meaningful. It DOES SO TOO. The argument is attempting to prove that two sets have different sizes. It therefore (for the purposes of of indirect proof) BEGINS BY ASSUMING THAT THEY ARE *THE SAME* size. That sameness MAKES THE LIST SQUARE. > In article > <db3f53df-2daa-4303-89a9-ad35f898a10b@m36g2000hse.googlegroups.com>, [quoted text clipped - 28 lines] > Cantor's argument does not require a square list, so it's not clear why > our failure to do so it meaningful. It does require that each listed string is endless, or can be extended endlessly with null characters. > The statements you make in your argument are true, but they don't apply > to Cantor's proof. That's the fundamental flaw in your argument. > > Alan >> In article >> <db3f53df-2daa-4303-89a9-ad35f898a10b@m36g2000hse.googlegroups.com>, [quoted text clipped - 31 lines] >It does require that each listed string is endless, or can be extended >endlessly with null characters. Sure, but "sufficiently infinite in both directions" seems like stetching the definition of "square" rather far. Anyway, if the infinite list isn't square then doesn't that prove the point just as well (which is a horribly informal argument, but that's what you get for talking about "square lists")? Alan SignatureDefendit numerus David R Tribble wrote: >> But your lists do /not/ contain their own anti-diagonal elements. george wrote: > No, stop; you're being completely uncharitable. > These lists ARE NOT SQUARE, and he CONCEDES this. > So YOU Have to concede that SOME sort of TWEAKING of the > MEANING of "diagonal" MUST be relevant. Tweaking the list so that it is "square" and thus its anti-diagonal is no longer constructable makes the whole exercise irrelevant to Cantor's proof. The resulting "tweaked" list may be of some interest, perhaps, but it can't lead to any conclusions about Cantor's list. [I thought you intended your response to be funny, but apparently you didn't.] In article <ccf5c734-cb8d-408d-ad61-10b739ecefef@e53g2000hsa.googlegroups.com>, > This is a disproof of the "diagonal argument" by induction It isn't, so [SNIP}. > Consider the inductive sequence of lists (a formal definition can be > found in [1]): [quoted text clipped - 3 lines] > 0 > 1 Is the same as: 00 10 The constructed number is 11, which is not in your list. > Over finite induction we see that every list is simply not square, and > that any anti-diagonal at each step, and actually any combination of > digits at each step, must be in the list for that step. [] What you just showed was that "the set of all numbers which have a terminating binary representation" (i.e. all numbers p/2^n for integer p, n) is countable. It's also important to remember that there is a distinction between "any finite number N" and "infinity." > > Consider the inductive sequence of lists (a formal definition can be > > found in [1]): [quoted text clipped - 8 lines] > 00 > 10 It is not. And you might even translate that to proper mathematical definitions, and still that will have nothing to do with the definitions given in the OP and, a fortiori, won't show any flaw in the given proof. -LV >>> Consider the inductive sequence of lists (a formal definition can be >>> found in [1]): [quoted text clipped - 11 lines] > definitions given in the OP and, a fortiori, won't show any flaw in > the given proof. (I'm assuming these are all binary decimals of the form 0.*) Question 1: Are the number represented by the binary decimal `0.1' and `0.10' the same? Question 2: If a proposition holds for some value of x, and x is equivalent to y, does the proposition hold for y? The answer to these questions are both "yes." If you disagree with either, you really are lacking in the foundations. Since 0.1 = 0.10, that means that whatever holds true for 0.1 must hold true for 0.10. It may be convenient for you to limit your lists of numbers to only those whose binary decimal places terminate within N places, but those lists don't come close to representing real numbers. Another way of looking at it is this: if your proof would hold true, that means that there must exist a natural number N such that every real number terminates within N (binary) decimal places, which means that every real number is of the form p/2^N (p being some natural number). Naturally, that is absurd. > >>> Consider the inductive sequence of lists (a formal definition can be > >>> found in [1]): [quoted text clipped - 13 lines] > > (I'm assuming these are all binary decimals of the form 0.*) Your (otherwise mathematically undefined) assumptions must be and are wrong in many ways. Indeed, if from an assumption you get the form of a reductio ad absurdum, what would you deduce? -LV > Question 1: Are the number represented by the binary decimal `0.1' and > `0.10' the same? [quoted text clipped - 15 lines] > every real number is of the form p/2^N (p being some natural number). > Naturally, that is absurd. >> (I'm assuming these are all binary decimals of the form 0.*) > > Your (otherwise mathematically undefined) assumptions must be and are > wrong in many ways. Indeed, if from an assumption you get the form of > a reductio ad absurdum, what would you deduce? So I erred in my assumption? What do the strings `0', `1', `00', `01', etc., mean, then? In article <617484c3-f1ec-484b-924c-e30bb93bd028@x35g2000hsb.googlegroups.com>, > > >>> Consider the inductive sequence of lists (a formal definition can be > > >>> found in [1]): [quoted text clipped - 16 lines] > Your (otherwise mathematically undefined) assumptions must be and are > wrong in many ways. Actually those assumptions are right in all ways relevant to this discussion. In article <e7cacd9a-d1b2-4881-b1d8-521d6cd86a0e@q9g2000hsb.googlegroups.com>, > > > Consider the inductive sequence of lists (a formal definition can be > > > found in [1]): [quoted text clipped - 13 lines] > definitions given in the OP and, a fortiori, won't show any flaw in > the given proof. It certainly does no show any flaws in Cantor's original diagonal proof, nor do any of the follow ups in this string. > Over finite induction we see that every list is simply not square, and > that any anti-diagonal at each step, and actually any combination of > digits at each step, must be in the list for that step. [] Exactly. This almost is A PROOF OF the diagonal lemma, as OPPOSED to a refutation of it: There are MORE than n bit-strings of length n, for every n. > In the limit case, There simply IS NO "limit case". There is no DEFINITION, in this context, of what a limit EVEN IS. There IS, as OPPOSED to the "limit" case, the INFINITE case. Since infinity IS NOT a natural number, and since induction ONLY Proves things (here) about (all) FINITE NATURAL numbers, and proves NOTHING WHATSOEVER about ANY infinite ANYthing, you simply have nowhere to go with this. > the length of the string may still be incomparable > to the length of the list, Why "still"?? In the finite case, the length (you should say width) of the string is NOT incomparable to the length of the list! They are EASILY compared, and WHENEVER you compare them, the result of that comparison IS ALWAYS that the length of the list is GREATER than the width of the bit-string! Specifically, width = lg length and length= xp width > but this -I suppose- shall be stated in > terms of an impossible bijection, When any set is of a different size from any other set, it is impossible to have a bijection between them, yes. > because the diagonal argument in > itself is (if I am not too mistaken) apparently and provably invalid, You are WAY too mistaken. > the above just being yet another disproof, The above IS NOT a disproof of the diagonal argument. > namely a counter-example > given by the simplest inductive construction. The above IS NOT a counter-example The diagonal argument IS CALLED that because it is about lists THAT HAVE diagonals because THEY ARE square. Since the above argument is about lists that ARE NOT square, it is simply not relevant to the issue. The whole point of the diagonal argument is that no SQUARE list contains its own anti-diagonal. If you want the list to contain its own anti-diagonal then it HAS to be LONGER than it is wide, and in that case, even clarifying a MEANING for "diagonal" requires stretching the concepts. The point about Cantor's theorem is simply to confirm that the list of subsets (or bit strings) IS LONGER than the width of the bit- string, i.e., bigger than the size of the underlying set. Your induction CONFIRMS that. It is NOT a COUNTER-example! > > Over finite induction we see that every list is simply not square, and > > that any anti-diagonal at each step, and actually any combination of [quoted text clipped - 3 lines] > as OPPOSED to a refutation of it: There are MORE than n bit-strings > of length n, for every n. It is not a proof of the diagonal lemma, it is what it is: a disproof of the diagonal argument by means of a counter-example. > > In the limit case, > > There simply IS NO "limit case". The proof ends at the "[]". -LV In article <2a4993f9-62ac-465a-82f8-fe8ef24f1f7a@m44g2000hsc.googlegroups.com>, > > > Over finite induction we see that every list is simply not square, and > > > that any anti-diagonal at each step, and actually any combination of [quoted text clipped - 6 lines] > It is not a proof of the diagonal lemma, it is what it is: a disproof > of the diagonal argument by means of a counter-example. Not according to any standard form of logic. Cantor says that for any list of strings (without any restrictions on string length) there is a string (of unspecified length) not in the list. So given any list with finite strings in it, every infinite string is different from all finite strings, and LV's arguments based on finite strings go down the tubes. > > > In the limit case, > > > > There simply IS NO "limit case". > > The proof ends at the "[]". It hasn't even started by the []. > The diagonal argument IS CALLED that because it is about lists THAT > HAVE > diagonals because THEY ARE square. Squares have FOUR EQUAL LENGTH SIDES. You seem to be implying that the diagonal argument only applies to lists that have RIGHT SIDES and BOTTOMS and are therefore filled with FINITE LENGTH strings. > > The diagonal argument IS CALLED that because it is about lists THAT > > HAVE [quoted text clipped - 3 lines] > the diagonal argument only applies to lists that have RIGHT SIDES and > BOTTOMS and are therefore filled with FINITE LENGTH strings. People talking to themselves? -LV ... > People talking to themselves? Why do you think that <geo_cant@yahoo.com> is the same as <greeneg@email.unc.edu>? Signaturedik t. winter, cwi, kruislaan 413, 1098 sj amsterdam, nederland, +31205924131 home: bovenover 215, 1025 jn amsterdam, nederland; http://www.cwi.nl/~dik/ |
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