Stability Theory

Abstract

Stability theory is a branch of mathematical logic which arose from classical model theory in the 1970's. It is a dividing line on the class of first-order theories, where a theory is stable if (in a certain size of domain) there are not too many types. It has proven useful in both model theory and algebra, but is difficult to understand due to the lack of easy textbooks. In this thesis, which is intended as lecture notes accessible for those with a basic knowledge of model theory, we introduce a number of concepts in this field. We consider various different definitions of stability, several rank functions, forking and forking independence, as well a few other dividing lines on the class of first-order theories. We conclude with a brief look at an application of stability theory to algebraic geometry. Exercises are scattered throughout the thesis, and although an appendix with solutions exists, this will not be made public (for now).