Godel's Theorem and Model Theory - sci.logic

Message from discussion Godel's Theorem and Model Theory

george

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More options Jul 26 2007, 8:43 am

Newsgroups: sci.logic

From: george <gree...@cs.unc.edu>

Date: Thu, 26 Jul 2007 12:43:58 -0700

Local: Thurs, Jul 26 2007 8:43 am

Subject: Re: Godel's Theorem and Model Theory

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> On Jul 26, 7:08 am, stevendaryl3...@yahoo.com (Daryl McCullough)
> > Well, the key statement that is wrong is the
> > claim "...this leads to a contradiction". No,
> > it doesn't. I can't say what's wrong with your
> > reasoning since you didn't say why you think it
> > leads to a contradiction.

On Jul 26, 3:25 pm, Newberry <newberr...@gmail.com> wrote:

> We have derived G,

Not in PA you haven't.
The reason why G is not derivable is that
it is NOT EVEN TRUE, in SOME models of PA.

> which says about itself that it is not derivable

NOthing "says anything about" itself, inherently.
Godel numbers are (almost) arbitrary.
Godel sentences (and consistency sentences)
are true under some interpretations and false under
others. The one you want to privilege (the natural
numbers) is NOT even DEFINABLE AT ALL at in the
context (r.e.classical first-order theories) that "derivability"
is about.

Basically, by "we have derived G", you meant,
"we have used ZFC to prove the existence of a model
of PA in which G is true". The far more interesting question
is actually about the models where it is false.

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