Re: Godel's Theorem and Model Theory
- From: Chris Menzel <cmenzel@xxxxxxxxxxxxxxxxxxxx>
- Date: Fri, 3 Aug 2007 16:24:13 +0000 (UTC)
On 3 Aug 2007 07:09:57 -0700, Daryl McCullough <stevendaryl3016@xxxxxxxxx> said:
george says...
On Aug 2, 2:07 pm, stevendaryl3...@xxxxxxxxx (Daryl McCullough) wrote:
That's true, but you don't have to add any *false* statements
to PA to get a nonstandard model for PA. It is possible to get
a nonstandard model of the *complete* theory of arithmetic.
Well, that is a pretty lame use of "non-standard", if it is going to
be elementarily equivalent to the standard.
Well, the truth is sometimes lame, I guess.
Though surely not in this case. It would be far lamer to use "standard"
in such a way that models of an order type other than omega, even
nondenumerable models, could count. It is (as Daryl would be the first
to agree) the structure of a model, not the first-order sentences that
are true in it, that makes it standard or not.
The usual meaning of "nonstandard model of arithmetic" is a model that
is not isomorphic to the standard model.
For good reason.
.
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