A Survey on Canonical Bases in Simple Theories

Abstract

Introduction The class of simple first order theories has recently been intensively studied. Simple theories were introduced by Shelah, who attempted to find a strictly broader class than the class of stable theories, in which forking is still reasonably well-behaved. After some years of neglect, the present author proved, in his doctoral thesis, that forking satisfies almost all the basic properties which hold in stable theories. In fact, a large amount of the machinery of stability theory, invented by Shelah, is valid in the broader class of simple theories. All stable theories are simple, but there are also simple unstable theories, such as the theory of the random graph. Many of the theories of particular algebraic structures which have been studied recently (pseudofinite fields, algebraically closed fields with a generic automorphism, smoothly approximable structures turn out to be simple unstable, too. Since a general survey of simple theories written by the author and A. Pillay appeared in the Bulletin of Symbolic Logic, here we shall focus on the notion of canonical base and Zilber's theorem in the context of simple theories. Roughly speaking, Zilber's theorem (Theorem 4.4) on a strongly minimal set says that any nontrivial ω-categorical, strongly minimal set is essentially a vector space over a finite field. Zilber used the notion of canonical base (for stable theories) significantly in the proof of this theorem.