Building Models by Games

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CUP Archive, May 2, 1985 - Mathematics - 311 pages
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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.
 

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Contents

GAMES AND FORCING
17
EXISTENTIAL CLOSURE
35
CHAOS OR REGIMENTATION
83
CLASSICAL LANGUAGES
132
PROPER EXTENSIONS
170
GENERALISED QUANTIFIERS
211
LQ IN HIGHER CARDINALITIES
250
List of types of forcing
275
Bibliography
282
Index
303
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Page 292 - KUEKER, DW, Back-and-forth arguments and infinitary logics. In: Infinitary Logic: In Memoriam Carol Karp (ed. DW KUEKER). Lecture Notes in Mathematics 492 (1975), 17-71.

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