Re: Godel's Theorem and Model Theory

george says...

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.

Any denumerable model by definition can be put into
AN order that will match the standard.
The ordering of the elements is not inherent;
it is purely a matter of viewpoint.
If the models are elementarily equivalent then the sentences
of the theory are not even going to be able to express OR NOTICE
the ALLEGED (from the meta-view) difference in the order-type, so
it is simply not going to be relevant.

If the question is: Can a complete theory have
nonisomorphic models, then it would seem that
the fact the models are nonisomorphic is relevant.

That a complete first order theory can have two nonisomorphic
models (of the same cardinality) is a fact about models. Maybe
it's a lame fact, but it is a fact, and it answers the question
asked: Can the complete theory of arithmetic have a nonstandard
model? Yes, it can.

--
Daryl McCullough
Ithaca, NY

.

Relevant Pages

  • Re: Skolem Paradox Question
    ... cardinal kappa". ... This is basically meaningful only if both ... standard and nonstandard models can exist. ... elementarily equivalent to whatever they are elementarily ...
    (sci.logic)
  • Re: Godels Theorem and Model Theory
    ... in such a way that models of an order type other than omega, ... I explicitly excluded nondenumerable models. ... that makes it standard or not. ... If the models are elementarily equivalent then the sentences ...
    (sci.logic)
  • Re: non-standard model of PA
    ... elementarily equivalent to the standard model. ... Most non-standard models are going to fail to ...
    (sci.logic)