Model completeness for trivial, uncountably categorical theories of Morley rank 1

We use this theorem to derive the same corollaries for the theories covered by the theorem as were derived for the strongly minimal case in [2]. We also note that the theorem is in some senses optimal. Specifically we can easily construct trivial Morley Rank 1 theories which are not categorical and for which the conclusion of the theorem fails. Also Marker in [3] constructs trivial totally categorical theories of Morley Rank 2 which are not model complete after naming any set of constants. Throughout the ensuing sections we rely heavily on the exposition presented in [2], so some familiarity with this paper will help the reader follow the present work. We will use basic concepts from stability theory without comment, see [1] for background material in the subject. Since we are generally concerned with the situation where we have models M ⊆ N and are attempting to prove that M N we do not have the luxury of working in a universal domain C and assuming that all models are elementary substructures of this structure. This entails that some of our notation and terminology differs somewhat from most references in stability theory, namely we must be very careful to specify the ambient model for some of the notions. We establish the following two notational conventions to fix the meaning of some basic stability theoretic concepts in our context. In the ensuing we assume T is a stable theory.