Stability Theory and its Variants

Dimension theory plays a crucial technical role in stability theory and its relatives. The abstract dependence relations defined, although combinatorial in nature, often have surprising geometric meaning in particular cases. This article discusses several aspects of dimension theory, such as categoricity, strongly minimal sets, modularity and the Zil’ber principle, forking, simple theories, orthogonality and regular types and in the third, stability, definability of types, stable groups and 1-based groups. One of the achievements of the branch of model theory known as stability theory is the use of numerical invariants, dimensions, in a broad setting. In recent years, this dimension theory has been expanded to include the so-called simple theories. In this paper, I wish to give just a brief overview of the elements of this theory. In the first section, the special case of strongly minimal sets is considered. In the second section, the combinatorial definition of dividing is given and how it leads to a general independence relation is outlined. Only in the third section do stable theories appear and the theory surrounding them is developed there with an eye to other papers in this volume. 1. Strongly Minimal Sets Categorical Theories. One of the simplest questions one can ask about a first order theory is how many models it has of a given cardinality. If T is a countable theory with an infinite model then, by the Lowenheim–Skolem Theorem, it will have at least one model of every infinite power. The situation we will look at first is when a theory has exactly one model of some fixed power. Definition 1.1. A theory T is λ-categorical if T has exactly one model up to isomorphism of cardinality λ. T is said to be totally categorical if T is λ-categorical for every infinite cardinal λ. We will say that T is uncountably categorical if T is λ-categorical for all uncountable λ. Example 1.2. 1. The theory of a set in a language which has only equality is totally categorical.