## Building Models by GamesThis 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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3j formula A+(p algebraically closed arithmetic assume boolean algebra chain compactness theorem compiled structure condition conjunction lemma constants construction contains Corollary countable e.c. countable first-order language countable model countable set define distinct witnesses dom(A dom(B e.c. group e.c. model element h elementarily equivalent elementary extension end extension enforceable equivalent modulo existentially closed finite-generic models first-order theory following are equivalent formula i|i(x formulas of L(Q group G hence hyperenforceable hypergame hypersupport induction infinite cardinal integer isomorphism joint embedding property Keisler L-structure largeness property Lemma limit ordinal Math model theory nilpotent groups notion of forcing play prove q forces quantifier quantifier-free formula realised recursive regular cardinal relation symbol Saracino satisfies Axioms sentence of L(W sequence set of formulas Shelah Show strongly omits Suppose Symbolic Logic tuple uncountable cardinal uncountable set weak L(Q)-structure winning strategy word problem write X-compact

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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.