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.
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31 formula A+(p algebraically closed arithmetic Aronszajn tree assume boolean algebra chain compactness theorem compiled structure condition constants construction contains Corollary countable e.c. countable first—order language countable model countable set define distinct witnesses dom(A e.c. group e.c. model elementarily equivalent elementary extension end extension enforceable equivalent modulo existentially closed finite-generic models first—order theory following are equivalent formulas of L(Q group G hence hyperenforceable hypergame hypersupport induction infinite cardinal isomorphism joint embedding property Keisler L—structure L(Q)-structure largeness property Lema Lemma limit ordinal locally finite group Macintyre Math maximal 3—type model theory N2 group nilpotent groups notion of forcing play prove q forces quantifier quantifier—free realised recursive regular cardinal relation symbol sequence set of formulas Shelah Show strongly omits Suppose Symbolic Logic tuple uncountable cardinal uncountable set V2 theory weak L(Q)—structure winning strategy write