Building Models by Games

This volume presents research by algebraists and model theorists in accessible form for advanced undergraduates or beginning graduate students studying algebra, logic, or model theory. It introduces a general method for building infinite mathematical structures and surveys applications in algebra and model theory. A multi-step procedure, the method resembles a two-player game that continues indefinitely. This approach simplifies, motivates, and unifies a wide range of constructions.
Starting with an overview of basic model theory, the text examines a variety of algebraic applications, with detailed analyses of existentially closed groups of class 2. It describes the classical model-theoretic form of this method of construction, which is known as "omitting types," "forcing," or the "Henkin-Orey theorem," The final chapters are more specialized, discussing how the idea can be used to build uncountable structures. Applications include completeness for Magidor-Malitz quantifiers, Shelah's recent and sophisticated omitting types theorem for L(Q), and applications to Boolean algebras and models of arithmetic. More than 160 exercises range from elementary drills to research-related items, with further information and examples.