FOM: second-order logic is a myth
Kevin Davey
kedst13+ at pitt.edu
Thu Mar 18 20:32:25 EST 1999
Regarding the following fragment of an exchange, quoted from a post of
Randall Holmes:
>Black and Simpson had the following exchange:
>
> > second-order logic is an essential tool for the *expression* of
> > mathematical theories. It can do this, and first-order logic
> > can't, ...
>
>That statement is indeed very odd, if not absurd, because it overlooks
>the fact that mathematics is formalizable in ZFC.
>
>I comment:
>
>I find it hard to believe that Simpson doesn't understand what Black
>means here. It is possible to formalize mathematics in ZFC in one
>sense, and not possible in another sense. It is possible to prove
>many facts that we believe about the mathematical world by using
>familiar representations of this world in ZFC. But it is not possible
>to prove certain other facts that we do believe in this language: for
>example, it is not possible to prove Con(ZFC), which can be understood
>as a fact about the natural numbers.
>
I just don't get Holmes' view here. Surely it -is- possible to express
Con(ZFC) in the language of set theory. Expression and proof are two
different things.
The best I can do is as follows. To argue that it is not the case that all
mathematics is formalizable in ZFC, it is proposed to establish that:
a) we have an -argument- for Con(ZFC), and
b) this argument cannot be formalized in ZFC.
Thankfully, b) is already taken care of by Godel. But why should I accept
a)? Of course, many of us have the -belief- that Con(ZFC) (and a sacred few
of us declare to have the -intuition- that Con(ZFC)), but unless this
somehow amounts to an -argument- for Con(ZFC), I don't see how one can
mobilize incompleteness to give a case against the expressive completeness
of 1st order logic.
Kevin Davey,
Department of Philosophy,
University of Pittsburgh.
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