Re: A question on GIT.

From: Mike Oliver (mike_lists_at_verizon.net)
Date: 09/08/04 

Date: Tue, 07 Sep 2004 23:52:35 -0500

namducnguyen wrote:

> Not knowing the complete proof of GIT, I guess I'm a little puzzled with a
> statement like "G(PA) is true in the standard model". I mean there seems to
> be a circularity in such statement:
>
> 1. We couldn't trust intuition [of arithmetic (infinity)], because of
> the many
> paradoxes about infinity.
> 2. So we tried to formalize out intuition, say, by using FOL framework.
> 3. But FOL framework is built principally on our _intuitive knowledge_ of
> arithmetic [which seems to be in conflict with 1.]
> 4. And then GIT states that arithmetic formalization is incomplete, and it
> states so using primarily *intuitive and incomplete* knowledge of
> standard
> model.
>
> I mean, it seems like we use incomplete intuition about infinity to
> prove that
> we couldn't completely trust our intuition about infinity!
>
> Naturally I don't have any intention to doubt GIT. It's just the
> seemingly circularity "bothers" me a little bit.

OK, so this *would* be circular, except that one of the
*hypotheses* of the theorem is that PA is consistent. If
PA is not consistent, then G(PA) is false, and PA is complete
(in a trivial sense: it proves every statement of the
language of arithmetic, and also refutes every such statement).

BTW Goedel's original result actually assumes the stronger
hypothesis that PA is omega-consistent; Rosser showed how
to weaken it to merely consistent.

Note that if PA is not consistent, then it doesn't have
any models at all.

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