Re: THIS STATEMENT HAS NO PROOF IN ANY SYSTEM = true or false?
poopdeville_at_gmail.com
Date: 02/07/05
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Date: 6 Feb 2005 18:26:18 -0800
tchow@lsa.umich.edu wrote:
> In article <vcbsm4cbmhd.fsf@beta19.sm.ltu.se>,
> Torkel Franzen <torkel@sm.luth.se> wrote:
> >poopdeville@gmail.com writes:
> >
> >>in the sense that if one makes an arithmetical claim, it is assumed
> >>to be formalizable in PA unless the speaker says he is referring
> >>to a different form.
> >
> > What do you mean by "formalizing a claim" in PA? Expressing it in
> >the language of PA? Proving it in PA? Most people, and indeed most
> >mathematicians, have no idea how to formalize anything in PA. What
> >is PA supposed to have to do with their thinking and talking about
> >arithmetic?
>
> Right. I think it would be a useful exercise for 'cid 'ooh to try to
> formalize something nontrivial, like the fundamental theorem of
arithmetic,
> in the first-order language of arithmetic. It's easy to make
fundamental
> mistakes when spouting generalities about formalization, unless one
has
> actually gone through the process in enough detail to understand all
the
> issues involved.
>
Well, if abbreviations (such as "n" for "s n-times(s)" and the like)
are allowed, expressing the fundamental theorem of arithmetic is
trivial. Of course, now you can object that we're using a proper
superset of the "normal language" of PA.
> For starters, professional number theorists spend a lot of their time
> talking about all kinds of arcane entities such as schemes over
finite
> fields, adeles/ideles, analytic continuations of L-functions, and
> whatnot. All these are soaked through and through with set-theoretic
> presuppositions.
Sure, and they if they're to prove anything about the naturals with
such methods, they must have some structure (in the non-technical
sense) that shows that these technologies bear on the natural numbers.
I would venture to guess that these structures need and in fact use
support (in the form of an axiomatization or model). It gets hairy
fast, I'll admit.
OK, you've made several great points I'm going to have to think about.
I still think my idea has merit, but I feel that I'm missing a lot of
nuance in what you've said (not here, but in previous posts), and that
I don't possess enough technical knowledge to support my philosophical
claims. So I'm going to study. Thanks for your criticism, comments,
complaints, conversation. (This goes to everyone else who participated
in "my" subthread, too. :-)
'cid 'ooh
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