A New Omitting Types Theorem

An omitting types theorem for countable models of superstable theories containing an infinite set of indiscernibles is proved. Various corollaries and applications are given. 1. One of the most useful ways that the model theorist has for controlling models is to attempt to specify the element types that models realize and omit. In this paper we present an omitting types theorem which seems well suited to applications to superstable theories, and whose proof relies on techniques from stability theory. For co-stable theories, the existence of prime models over arbitrary subsets allows one to omit many types, but the existence of such models is not guaranteed in superstable theories. It is hoped that the technique developed here allows one to obtain some of the consequences of the existence of prime models over arbitrary subsets when such models are not available. In §2, the basic omitting types theorem is proved and some counterexamples to various strengthenings are given. Some applications and corollaries of the theorem are presented in §3. The notation used throughout is standard. 2. We assume the reader is familiar with some of the basic notions of stability theory. Any unexplained terms may, of course, be found in [4, III, §§1-3]. The main result of this section is Theorem 2.1. Let T be superstable and 911 a countable model of T containing an infinite set of indiscernibles. If p(x, m) is a type with a finite sequence of parameters m E 911 that is omitted in 911, then there is an 91^ 911 that also omits p(x, m). Before proving Theorem 2.1, we shall have to avail ourselves of some preliminaries. All results may be found in [4, III, §§1-3]. Definition 2.2. Let T be stable, C E 9H t= T and I E 911 be an infinite sequence (hence set) of indiscernibles. The average type of I over C, denoted by Av(7, C), is [ N0}. The lemma below justifies the terminology employed in the definition. Lemma 2.3. For stable T, the average type of I over C, for any I and C, is a consistent and complete set of formulas. Received by the editors January 19, 1983. 1980 Mathematics Subject Classification. Primary 02H05, 02H13. 1 Partially supported by a grant from NSERC of Canada. ©1983 American Mathematical Society 0002-9939/83 $1.00 + $.25 per page