[FOM] explicit construction; choice and model theory

[Harvey Friedman]

I want to correct the mathematical material at the end of my posting 
10:57PM 7/2/04.

I wrote

>I can give a formulation of this that MIGHT come within what you are 
>calling model theory. This alternative formulation still fits 
>squarely into my perspective.
>
>DEFINITION. A model theoretic site consists of a nonempty set D 
>together with a collection S of relations of several variables on D 
>which is closed under first order definability. I.e., any relation 
>that can be defined from finitely many relations in S, lies in S.
>
>PROBLEM. Let (D,S) be a model theoretic site. Give a necessary and 
>sufficient condition for the following to hold. Every sentence with 
>an infinite model has a model with domain D whose relations (and 
>graphs of its functions) lie in S.
>
>ANSWER???? If and only if D is infinite, S has (the graph of) a 
>one-one map from DxD into D, and S has a linear ordering of D.
>
>ANSWER. If and only if S has a model of (a weak fragment of) PA.
>
PROBLEM. Give a simple direct mathematical answer, not in terms of a 
formal system like PA.

************************

The ANSWER???? and the ANSWER are both wrong. There cannot be such an 
answer. Here is a correct answer - though it is far too 
metamathematical an answer.

THEOREM. Let (D,S) be a model theoretic site. The following are equivalent.
i) every sentence that has an infinite model has a model with domain 
D whose relations (and graphs of functions) lie in S;
ii) For every true Pi-0-1 sentence phi, there is a model of PA + phi 
with domain D, in S.

Harvey Friedman