Decidable prime models

Abstract

A set S of types over a theory T is strongly free if for all subsets X [strict subset] S, there is a countable model of T which realizes X and omits S\X. Throughout, all theories are assumed complete and consistent unless otherwise stated. Theorem 1 If all strongly free sets of types over a recursive theory T are finite, then T has a decidable prime model. Definition 2 A model is decidable if it is isomorphic to a model whose elementary diagram is recursive (technically speaking, this just means the model has a decidable presentation. Throughout this paper, however, we will just say the model is decidable} A classical result in model theory is that any theory with less than 2No many countable models must have a prime model. Our theorem gives an effective extension of this result: Corollary 3 If a countable theory T has less than 2No many countable models, then there is a prime model of T decidable in T.