Trivial pursuit: Remarks on the main gap

Abstract

A dependence relation is trivial when a single point which depends on a pair of independent points depends on just one of them. In this paper we survey some of the salient features of this simple but important concept and then discuss some of the major applications of it in stability theory. Although the actual study of trivial types is rather trivial the pursuit of them is an exciting game. One reward for playing this game is a quicker calculation of the number of models of an o-stable theory without the dimensional order property. This paper amalgamates several sections from Baldwin’s forthcoming book on stability theory. Our discussion of trivial types clarifies the proof of the result on the spectrum problem and allows for several improvements. The improvements in the proof of the main gap results came about during long discussions between the authors and later with Buechler. Shelah has shown that each theory which is not superstable has the maximal number of models in each uncountable power. Moreover, superstable theories with the dimensioi_al order property have the maximal number of models in each uncountable power [ll, 71. We assign an invariant to each theory without the dimensional order property, the depth of T, which bounds the complexity of a tree which decomposes a model of T. If this depth is infinite and X, > X,, I(&, K) was calculated by Harrington and Makkai [7] following Shelah [ll]. We extend that result here by requiring in the infinite depth case only that X, 3 xz. The proof of the finite depth case was given by Saffe [lo]. The methods of the proof here are extended to that case in Chapter XVIII of [l]. We also answer there a question of Saffe by giving a classification of the countable o-stable theories with finite depth which accounts for the spectrum in both countable and uncountable powers. Section 1 of this paper contains some technical results about trivial types. Section 2 describes the properties of the trees which form skeletons of theories with ndop, counts the number of such trees, and thus describes an