Building Models by Games

This book introduces a general method for building infinite mathematical structures, and surveys its applications in algebra and model theory.

The basic idea behind the method is to build a structure by a procedure with infinitely many steps, similar to a game between two players that goes on indefinitely.

The approach is new and helps to simplify, motivate and unify a wide range of constructions that were previously carried out separately and by ad hoc methods.

The first chapter provides a resume of basic model theory.

A wide variety of algebraic applications are studied, with detailed analyses of existentially closed groups of class 2.

Another chapter describes the classical model-theoretic form of this method -of construction, which is known variously as 'omitting types', 'forcing' or the 'Henkin-Orey theorem'.

The last three chapters are more specialised and discuss how the same idea can be used to build uncountable structures.

Applications include completeness for Magidor-Malitz quantifiers, and Shelah's recent and sophisticated omitting types theorem for L(Q).

There are also applications to Bdolean algebras and models of arithmetic.