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Categoricity

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Overview

Modern model theory began with Morley's categoricity theorem: A countable first-order theory that has a unique (up to isomorphism) model in one uncountable cardinal (i.e., is categorical in cardinality) if and only if the same holds in all uncountable cardinals. Over the last 35 years Shelah made great strides in extending this result to infinitary logic, where the basic tool of compactness fails. He invented the notion of an Abstract Elementary Class to give a unifying semantic account of theories in first-order, infinitary logic and with some generalized quantifiers. Zilber developed similar techniques of infinitary model theory to study complex exponentiation. This book provides the first unified and systematic exposition of this work. The many examples stretch from pure model theory to module theory and covers of Abelian varieties. Assuming only a first course in model theory, the book expounds eventual categoricity results (for classes with amalgamation) and categoricity in excellent classes. Such crucial tools as Ehrenfeucht-Mostowski models, Galois types, tameness, omitting-types theorems, multi-dimensional amalgamation, atomic types, good sets, weak diamonds, and excellent classes are developed completely and methodically. The (occasional) reliance on extensions of basic set theory is clearly laid out. The book concludes with a set of open problems.

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Product Details

  • ISBN-13: 9780821848937
  • Publisher: American Mathematical Society
  • Publication date: 7/24/2009
  • Series: University Lecture Series , #50
  • Pages: 235
  • Product dimensions: 7.00 (w) x 9.70 (h) x 0.50 (d)

Table of Contents

Pt. 1 Quasiminimal Excellence and Complex Exponentiation 1

Ch. 1 Combinatorial Geometries and Infinitary Logics 3

Ch. 2 Abstract Quasiminimality 7

Ch. 3 Covers of the Multiplicative Group of C 17

Pt. 2 Abstract Elementary Classes 25

Ch. 4 Abstract Elementary Classes 27

Ch. 5 Two Basic Results about L[subscript w1,w](Q) 39

Ch. 6 Categoricity Implies Completeness 45

Pt. 3 Abstract Elementary Classes with Arbitrarily Large Models 63

Ch. 8 Galois types, Saturation, and Stability 67

Ch. 9 Brimful Models 73

Ch. 10 Special, Limit and Saturated Models 75

Ch. 11 Locality and Tameness 83

Ch. 12 Splitting and Minimality 91

Ch. 13 Upward Categoricity Transfer 99

Ch. 14 Omitting Types and Downward Categoricity 105

Ch. 15 Unions of Saturated Models 113

Ch. 16 Life without Amalgamation 119

Ch. 17 Amalgamation and Few Models 125

Pt. 4 Categoricity in L[subscript w1, w] 133

Ch. 18 Atomic AEC 137

Ch. 19 Independence in w-stable Classes 143

Ch. 20 Good Systems 151

Ch. 21 Excellence Goes Up 159

Ch. 22 Very Few Models Implies Excellence 165

Ch. 23 Very Few Models Implies Amalgamation over Pairs 173

Ch. 24 Excellence and *-Excellence 179

Ch. 25 Quasiminimal Sets and Categoricity Transfer 185

Ch. 26 Demystifying Non-excellence 193

Appendix A Morley's Omitting Types Theorem 205

Appendix B Omitting Types in Uncountable Models 211

Appendix C Weak Diamonds 217

Appendix D Problems 223

Bibliography 227

Index 233

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