Definability in low simple theories

Abstract

In 1978 Shelah introduced a new class of theories, called simple (see [Shi]) which properly contained the class of stable theories. Shelah generalized part of the theory of forking to the simple context. After approximately 15 years of neglecting the general theory (although there were works by Hrushovski on finite rank with a definability assumption, as well as deep results on specific simple theories by Cherlin, Hrushovski, Chazidakis, Macintyre and Van den dries, see [CH], [CMV], [HP1], [HP2], [ChH]) there was a breakthrough, initiated with the work of Kim ([K1]). Kim proved that almost all the technical machinery of forking developed in the stable context could be generalized to simple case. However, the theory of multiplicity (i.e., the description of the (bounded) set of non forking extensions of a given complete type) no longer holds in the context of simple theories. Indeed, by contrast to simple theories, stable theories share a strong amalgamation property of types, namely ifpandqare two “free” complete extensions over a superset ofA, and there is no finite equivalence relation overAwhich separates them, then the conjunction ofpandqis consistent (and even free overA.) In [KP] Kim and Pillay proved a weak version of this property for any simple theory, namely “the Independence Theorem for Lascar strong types”. This was a weaker version both because of the requirement that the sets of parameters of the types be mutually independent, as well as the use of Lascar strong types instead of the usual strong types. A very fundamental and interesting problem is whether the independence theorem can be proved for any simple theory, using only the usual strong types. In 1997 Buechler proved ([Bu]) the strong-type version of the independence theorem for an important subclass of simple theories, namely the class of low theories (which includes the class of stable theories and the class of supersimple theories of finite D-rank.)