Instability and Theories with Few Models

Some results are obtained concerning n(T), the number of countable models up to isomorphism, of a countable complete first order theory T. It is first proved that if n(T) = 3 and T has a tight prime model, then T is unstable. Secondly, it is proved that if n(T) is finite and more than one, and T has few links, then T is unstable. Lastly we show that if T has an algebraic model and has few links, then n(T) is infinite. We work with countable complete theories. Under certain assumptions which simplify the structure of a theory, we prove that such a theory with few countable models (finitely many and more than one, or in some cases exactly three) is unstable. The first condition is that every countable model of T has an elementary substructure which is prime and contains every other prime elementary substructure. This condition is first examined and related to other properties of prime models. Then it is shown that if such a theory has exactly three countable models then it is unstable. Secondly we look at the condition that a theory has 'few links'. This was introduced by Benda in [1], and roughly speaking says that given any two types there are few ways of relating them in a nonprincipal way. Benda proved that a theory with few links and few countable models has at least two countable universal models. We show that a theory with few links and a finite number of countable models is unstable. (A weaker property than few links is actually needed.) Actually we show that such a theory 'says' that there is a dense ordering, and we conclude that a theory with few links and an algebraic model has infinitely many countable models. 1. Tight prime models. We assume here that T is atomic (the principal types are dense in the space of (complete) types). We know that such a theory has a prime model (elementarily embeddable in all models of T) which is unique up to isomorphism. Definition 1. Let M be the prime model of T. Then M is said to be tight if for any countable model A^ of T, the set of elementary substructures of N which are isomorphic to M, has a greatest element under the ordering by inclusion. We recall that a minimal model is a model with no proper elementary substructure. If T has a minimal model then it is unique and prime. Received by the editors April 12, 1979. 1980 Mathematics Subject Classification. Primary 03C15, 03C45; Secondary 03C50.