Categoricity and Ranks

In this paper we investigate the connections between categoricity and ranks. We use stability theory to prove some old and new results. ?0. Introduction. In this paper we shall investigate the connections between ranks and categoricity. Ranks have played a central role in stability theory from the very beginning. An almost complete treatment of ranks of (not necessarily complete) types can be found in Chapter II of Shelah's book [Shl]. Another rank that has been proved to be very useful has been introduced by Lascar in [Lsl]. Almost immediately after the definitions people realized that there are intimate connections between categoricity and ranks-for example, categoricity implies that certain ranks coincide in this special case which do not coincide in general or that in categorical theories necessarily some rank is finite (see mainly [Bal], [Lsl] and [La]). This paper is intended to be a collection of such results; so most results presented here are not new. So if the reader is only interested in new results he should only read Theorems 3.6 and 5.2. But if he is interested in new proofs as well he can read all the paper because we shall replace the first proofs, which were often very "ad hoc" by, (hopefully) simpler ones-so it may become clear why the theorems are true; and sometimes this leads to generalizations as well. But this requires knowledge of some machinery-the one needed here is clearly forking. So a successful reader will have a reasonable knowledge of [Shl]reasonable in this context means "at least including ?V.3". Especially we shall make use of the main theorems of ?IV.4. Another basic source is [LP], which should be familiar to the reader as well. For a better understanding we invite the reader to study [Ls2] and [Sf], where the material on regular types and related notions is developed from a different point of view. The real importance of these notions can only be seen by looking at the series [Sh2]. Our notation will be what has in our opinion become "standard" in this area, i.e. we shall follow [Shl] and [LP]. For example, every set and model under consideration is supposed to be part (respectively, an elementary submodel) of a big saturated model that is not to be mentioned. T(A) denotes the theory got by naming all elements of A (where A is the underlying set of A and A is not necessarily a model) Received October 12, 1983. ( 1984, Association for Symbolic Logic 0022-4812/84/4904-0030/$02.40