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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, Shelahs 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.

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