Re: Goedel incompl. theorems


Henrik wrote:
What do you mean with "But, what ordinary mathematicians use is
essentially 2nd-order logic without the formalism"? Is not most of
ordinary mathematics possible to formalize in ZFC, which is a first
order theory?

ZFC is not inherently a first-order theory.
"2nd-order ZFC" is NOT a contradiction in terms,
any more than "2nd-order PA" is.
The canonical first-order axiomatizations of both of these
theories have a first-order axiom-SCHEMA (in PA it is induction;
in ZFC it is replacement) that can be very naturally lifted
to 2nd order to give a 2nd-order version of "the same" axioms.

"Most" "ordinary" mathematics probably can be done
in first-order ZFC. But asking this question naturally
focuses people's attention on the things that can't be done
in first-order ZFC. Once you start focusing on those, it
becomes very uncouth to bar them as "unnatural".

The general meta-theory of first-order ZFC would of course
be the most obvious thing you can't treat in first-order ZFC.
Category theory, group theory, and permutation theory
in general also leap to mind, the point being that the ZFC translations
of all these you to unnaturally limit these to some particular
universe/domain/bounding-set.

.

Relevant Pages

  • Re: Goedel incompl. theorems
    ... essentially 2nd-order logic without the formalism"? ... ZFC is not inherently a first-order theory. ... in first-order ZFC. ...
    (sci.logic)
  • Re: axioms of mathematical logic
    ... First, within the theory of recursive functions, there is known a proper ... i base it on predicate logic and ZFC. ... I simply pointed to that there is naive usage of 'very' finite formalism not ... arithmetization. ...
    (sci.logic)
  • Re: Countable models of ZFC
    ... The whole POINT of its being a FORMALISM ... not be "referring to" ANYthing! ... constructing a model in ZFC as a consistency proof ... That "the axioms of ZFC force themselves upon us as true" ...
    (sci.logic)
  • Re: How do We Know that ZF is the Axiomatization that Proves everything Provable?
    ... For example, the axioms of PA prove facts about arithmetic, ... the same theorems. ... level formalism includes that and other sets ... So lets stick to FOL and ZFC, eh voila there are *already* some gaps. ...
    (sci.logic)
  • Re: 1-1/2+1/3-1/4+1/5-1/6+1/7
    ... the content within the formalism. ... checking whether a sequence is a proof in ZFC is completely ... mathematical proofs are a blend of the formal and informal. ... skill of a mathematician that once he has "figured things out in his ...
    (sci.math)
Quantcast