Recent results on simple first-order theories

Summary

This paper summarizes the main results of and some facts from with condensed proofs.

The study of simple theories began with Shelah's paper “Simple unstable theories” where he introduced a class of first order theories, he called simple, having D(p,Δ, k) rank. The class includes all stable theories and some unstable theories. His intention was to ask whether we can build a theory of simple theories analogous to stability theory.

Remarkable progress in the study of simple theories has been made very recently after Hrushovski and others developed notions of independence in specific unstable structures such as pseudo-finite fields, fields with an automorphism and smoothly approximable structures. The independence notion in each of these unstable structures behaves similarly to nonforking in stable structures. Hence we may ask naturally the following questions in connection with simple unstable theories.

1. (1) Are all unstable structures mentioned above simple?

2. (2) If so, then “what is the relation between the independence notion and non-forking?”

3. (3) Does any simple theory have a similar notion of independence?

It turns out that the answer to (1) is positive and the independence notion in each unstable structure above is exactly nonforking. Furthermore nonforking supplies a notion of independence to all simple unstable theories as well as to stable theories, primarily by the following theorem.