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.
Author(s): John T. Baldwin
Series: University Lecture Series
Publisher: American Mathematical Society
Year: 2009
Language: English
Pages: 248
Contents......Page 3
Introduction......Page 5
Part 1. Quasiminimal Excellence and Complex Exponentiation......Page 11
1.1 Combinatorial geometries......Page 13
1.2 Infinitary logic......Page 14
2. Abstract Quasiminimality......Page 17
3. Covers of the Multiplicative Group of C......Page 27
Part 2. Abstract Elementary Classes......Page 35
4. Abstract Elementary Classes......Page 37
5.1 Non-definability of well-order in L_{ω_1,ω}(Q)......Page 49
5.2 The number of models in ω_1......Page 51
6.1 Completeness......Page 55
6.2 Arbitrarily large models......Page 60
6.3 Few models in small cardinals......Page 62
6.4 Categoricity and completeness in L_{ω_1,ω}(Q)......Page 64
7. A Model in ℵ_2......Page 67
Part 3. Abstract Elementary Classes with Arbitrarily Large Models......Page 73
8. Galois Types, Saturation and Stability......Page 77
9. Brimful Models......Page 83
10. Special, Limit and Saturated Models......Page 85
11. Locality and Tameness......Page 93
12. Splitting and Minimality......Page 101
13. Upward Categoricity Transfer......Page 109
14. Omitting Types and Downward Categoricity......Page 115
15. Unions of Saturated Models......Page 123
16. Life without Amalgamation......Page 129
17. Amalgamation and Few Models......Page 135
Part 4. Categoricity in L_{ω_1,ω}(Q)......Page 143
18. Atomic AEC......Page 147
19. Independence in ω-stable Classes......Page 153
20. Good Systems......Page 161
21. Excellence Goes Up......Page 169
22. Very Few Models implies Excellence......Page 175
23. Very Few Models implies Amalgamation over Pairs......Page 183
24. Excellence and *-excellence......Page 189
25. Quasiminimal Sets and Categoricity Transfer......Page 195
26.1 The basic structure......Page 203
26.2 Solutions and categoricity......Page 206
26.3 Disjoint amalgamation for models of φ_k......Page 210
26.4 Tameness......Page 211
26.5 Instability and non-tameness......Page 212
Appendix A. Morley's Omitting Types Theorem......Page 215
Appendix B. Omitting Types in Uncountable Models......Page 221
Appendix C. Weak Diamonds......Page 227
Appendix D. Problems......Page 233
Bibliography......Page 237
Index......Page 243
Errata......Page 246
