Re: question about axiomatic set theory

NSA involves a similar kind of circularity, that I tried to describe in
my last posting - considering that the nonstandard model of N, for
example, is an enlargement of N, it is a construction that belongs
properly to logic. But ultraproducts are used in this construction -
the validity of the axioms of the enlargement is concluded from the
compactness theorem, a version which uses the axiom of choice in its
proof. Suppose we don't want the axiom of choice in the basic, informal
set "theory" that establishes logic. In that case, there is the same
"circularity involved in using sets when defining how logic works".

As I noted before, as a way out of this, you might say: ok, now we've
got logic from basic set theory; now we'll develop ZFC, and then apply
it back to logic, by (necessarily) considering logic as a set-theoretic
(not logical) structure, sort of like a category whose elements are in
the form (semmantic structure M, formal system F, correspondence
M-->L(F)). In that context, nonstandard "models" can be defined.

The problem is that this seems very an undesirable, messy solution - if
N* is a set-theoretic structure, it's not really an extension of the
same N as the one discussed two paragraphs ago.

This is not a problem in a lot of mathematics, but is only very
apparent when the object of mathematical inquiry involves mathematics
itself. In other words, mathematical logic is like philosophizing from
within the world and language of mathematics (the number 7 woke up one
day and started to ask "what am I?"), it is self-referring,
self-conscious so to speak. And on that line, to apply it's methods to
mathematics is analogous to applying philosophy to politics.

.

Relevant Pages

  • Re: Skolems Paradox and why is math the way it is?
    ... It depends on how stringent your criteria for "circularity" are. ... but the first part (formalizing set theory) was largely successful. ... the axioms as purely formal strings of symbols without any "meaning" ... "non-circularity," then circularity is unavoidable in mathematics. ...
    (sci.math)
  • Godels incompleteness theorm proven wrong
    ... either lead to paradox or end in paradox. ... axiom of reducibility but this axiom was rejected as being invalid by ... Thus just on this point Godel is invalid as by using an axiom most people ... Godel proved that mathematics was inconsistent ...
    (sci.math)
  • Godels incompleteness theorem proven invalid
    ... axiom 1V - axiom of reducibility- in his proof. ... either lead to paradox or end in paradox. ... Thus just on this point Godel is invalid as by using an axiom most people ... Godel proved that mathematics was inconsistent ...
    (sci.math)
  • Re: Why does Cantor a target for cranks?
    ... objects without having some sort of axiom to that effect. ... Has to have an axiomatic system infinite many axioms to be definite? ... My point is that, at least in in pure mathematics, one cannot conclude ... The axiomatic method does not in any way dispute what you have just ...
    (sci.math)
  • Re: Why does Cantor a target for cranks?
    ... objects without having some sort of axiom to that effect. ... Has to have an axiomatic system infinite many axioms to be definite? ... My point is that, at least in in pure mathematics, one cannot conclude ... The axiomatic method does not in any way dispute what you have just ...
    (sci.math)