Re: Godel's Theorem and Model Theory

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. The usual meaning of
"nonstandard model of arithmetic" is a model that is not isomorphic
to the standard model.

I did make an effort at PRIOR exclusion of the models that were too big.
There is a serious linguistic problem here in that the nonstandard
models necessarily contain elements that necessarily HAVE NO NAMES
in the standard language. You arguably DO have to add something like
a false existential (or its skolemization)
in order to be able to expand the language enough to be able to even
REFER to any of the nonstandard elements.

That's all true, but it is still the case that there exist nonstandard
models of the complete theory of arithmetic.

--
Daryl McCullough
Ithaca, NY

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