Building Models by Games

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Courier Corporation, 2006 - Mathematics - 318 pages
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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, 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.

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About the author (2006)

Ian Chiswell acheived a Ph.D. at the University of Michigan in 1973 on the Bass-Serre theory of groups acting on trees. After three years as a temporary lecturer at the University of Birmingham he moved back to Queen Mary, University of London in 1976. His teaching experience dates back to 1968
when he was a teaching fellow at the University of Michigan. He spent the academic year 1972-73 in Germany at the Ruhr-Universitaet Bochum. He has published a monograph on lamda-trees, which are generalisations of ordinary trees. His work has connections with mathematical logic, mainly via
non-standard free groups. Wilfrid Hodges achieved his DPhil at Oxford in 1970 for a thesis in model theory (mathematical logic). He has taught mathematics at London University for nearly forty years, first at Bedford College and then at Queen Mary, and also taught for visiting years in Los Angeles
and Boulder (USA). Besides this book, he has four other textbooks of logic in print, at levels ranging from popular to research. He has served as president of the British Logic Colloquium and the European Association for Logic, Language and Information, and as vice-president of the London
Mathematical Society.

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