# [FOM] explicit construction; choice and model theory

Harvey Friedman friedman at math.ohio-state.edu
Thu Jul 3 01:15:11 EDT 2003

```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