Math Forum / Mathematics / Mathematical Logic / February 2006

There is a proof of validity for every valid first-order sentence. Is
it possible to prove, in first-order logic itself, that a sentence is
not valid?
/ALiX

Are there Sigma_{n>1}^1 (Sigma-1-1) fragments of SOL that are proper
extensions of the Sigma_1^1 fragment? Are all of them?
Also, what's the deal with quantifier collapsing in arithmetic? E.g.
the formula (Ex)(Ey)A is equivalent to a (Ex)B formula. (Perhaps B is

I am in both an introductory logic and introductory discrete
mathematics course this semester. My textbooks are A Concise
Introduction to Logic by Patrick J. Hurly and Discrete Mathematics by
Dossey, Otto Spence and Vanden Eynden. The logic textbook uses the

Vladimir Voevodsky is giving lectures at Stanford on the
"homotopy lambda calculus":
http://math.stanford.edu/distinguished_voevodsky.htm
Can anyone report on what he said?

I have been able to show that the equivalent all but one of Peano's
Axioms can be derived from the fields axioms -- namely that no natural
number has a successor of 1. See:
http://www.dcproof.com/PeanoFieldAxioms.html

I call it a conjecture because Appel & Haken and R,S,S,T with their
computer caseloads of 1476 and 633 are not convincing as proof
arguments, because they are too long and rely on computers, and because
the numbers 1476 and 633 have no special significance to geometry. So

I need to post this to sci.physics also because the best example is
that of physics where if the Periodic Table of Elements had no smallest
atom, would atoms exist? If there was no smallest Element, would any
element exist? Obviously not.

I have never understood why set-exponentiation (like in Martin Loef's Type
Theory) is constructive.
I would discuss this topic here.
GL

It is clear to me how using Russell's predicate with the Axiom of
Abstraction leads to the false statement (contradiction)
Ey   y e y  <=>  ~  y e y
This is  corrected in the Axiom of Separation, introducing set z, which

In the ongoing discussions in the FOM newsgroup, Nik Weaver discusses
his version of predicativism, or what he calls "conceptualism". You can
see Weaver's work at his website
<http://www.math.wustl.edu/~nweaver/conceptualism.html> and also in the

I couldn't find the first mention of the Moebius theorem (at most
4 mutually adjacent regions in a planar map - ?), so I'm just starting
a hopefully brief topic.
The only mention of this Moebius theorem I could find by googling

My understanding is that the following is an inductively forceful argument:
 P) Most people are happy.
 C) Simon is happy.
But is the following a deductively valid argument?:

To the following questions (not addressed to myself) on the FOM board
"Do you endorse that standard [ZFC]? Or do you want to suggest that the
Annals of Mathematics and the Transactions of the AMS, etcetera, change
that standard?"

Research at the University of Cambridge supports the following ideas:
1) the increasing incidence of autism is partially linked to
analytically minded people finding each other at work - women are now
better entrenched in the engineering, computer programing, etc. fields.

Newbie here...
I really don't understand why the sixth instance below doesn't hold for
math but supposedly holds for logic... I'll list below what you guys
already know, but please try to provide an intuitive explanation as to