A counterexample in the theory of model companions

Abstract

In [7] we proved that (I) if T is a countable ℵ0-categorical theory without finite models then T has a model companion; and several people have observed that (II) if T is a countable theory without finite models which is ℵ1-categorical and forcingcomplete for infinite forcing (i.e., T= TF) then T is model-complete. It is natural to ask (1) whether in (I) we can replace ℵ0 by ℵ1; (2) whether in (II) we can replace TF by Tf; and (3) in connection with (II), whether the categoricity of the class of infinitely generic structures for a theory K in some or all infinite powers implies the existence of a model companion for K. The purpose of this note is to provide negative answers to (1), (2), and (3). Specifically, we will prove:Theorem. There exists a countable theory T such that(i) T has no finite models and is ℵ-categorical;(ii) T is forcing-complete for finite forcing, i.e., T = Tf;(iii) T has no model companion (i.e., in light of (ii), T is not model-complete);(iv) the class of infinitely generic structures for T is categorical in every infinite power;(v) every uncountable existentially complete structure for T is infinitely generic;(vi) there is, up to isomorphism, precisely one countable existentially complete model of Tf, and there are no uncountable e.c. models of Tf (in particular, there is just one countable finitely generic structure and there are no uncountable ones);(vii) there are precisely ℵ0isomorphism types of countable existentially complete structures for T.