Re: Godel's theorem is invalid?

Torkel Franzen wrote:
> "george" <greeneg@xxxxxxxxxx> writes:
>
> > > But there is a big difference between
> > >
> > > (1) For all n, we can prove P(n).
> > >
> > > and
> > >
> > > (2) We can prove "for all n, P(n)"
> >
> > Not if you don't know that there can be models
> > with things other than natural numbers in them, there
> > isn't.
>
> Why drag in the model theory of first order logic? It's irrelevant
> to the distinction between (1) and (2).

In classical logic, it is assumed that (1) is a meaningful assertion
with n taken successively as 0, 1, 2, etc., i.e., it is assumed that
one can run through "all" the natural numbers by thinking about them
one (or finitely many) at a time. I submit that this is false. A simple
induction shows that one cannot exhaust the natural numbers by listing
them one at a time.

In fact we can *know* that (1) is true, and hence a meaningful
assertion, if and only if:

(3) We can prove P(n) for an *arbitrary* natural number n, where
n is left in symbolic form.

Hence there *should* really be no distinction between (1) and (3), a
distinction which classical first-order logic dubiously tries to
maintain. By the rule of universal generalization, (3) *should be* the
same as (2), and so nonstandard models of arithmetic *ought* not to
exist. This is exactly what my proposed logic NAFL asserts. See the
following links:

http://philsci-archive.pitt.edu/archive/00001666
http://philsci-archive.pitt.edu/archive/00001923
http://arxiv.org/abs/math.LO/0506475

I am still wating for a substantial response from the academic
community on the above work.

Regards, RS

.

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