**University Lecture Series**

Volume: 50;
2009;
235 pp;
Softcover

MSC: Primary 03;

Print ISBN: 978-0-8218-4893-7

Product Code: ULECT/50

List Price: $58.00

Individual Member Price: $46.40

**Electronic ISBN: 978-1-4704-1645-4
Product Code: ULECT/50.E**

List Price: $58.00

Individual Member Price: $46.40

#### Supplemental Materials

# Categoricity

Share this page
*John T. Baldwin*

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.

#### Table of Contents

# Table of Contents

## Categoricity

- Cover Cover11 free
- Title page iii5 free
- Contents v7 free
- Introduction vii9 free
- Part I. Part 1. Quasiminimal excellence and complex exponentiation 115 free
- Combinatorial geometries and infinitary logics 317
- Abstract quasiminimality 721
- Covers of the multiplicative group of ℂ 1731
- Part II. Part 2. Abstract elementary classes 2539
- Abstract elementary classes 2741
- Two basic results about 𝐿_{𝜔₁,𝜔}(𝑄) 3953
- Categoricity implies completeness 4559
- A model in ℵ₂ 5771
- Part III. Part 3. Abstract elementary classes with arbitrarily large models 6377
- Galois types, saturation, and stability 6781
- Brimful models 7387
- Special, limit and saturated models 7589
- Locality and tameness 8397
- Splitting and minimality 91105
- Upward categoricity transfer 99113
- Omitting types and downward categoricity 105119
- Unions of saturated models 113127
- Life without amalgamation 119133
- Amalgamation and few models 125139
- Part IV. Part 4. Categoricity in 𝐿_{𝜔₁,𝜔} 133147
- Atomic AEC 137151
- Independence in 𝜔-stable classes 143157
- Good systems 151165
- Excellence goes up 159173
- Very few models implies excellence 165179
- Very few models implies amalgamation over pairs 173187
- Excellence and *-excellence 179193
- Quasiminimal sets and categoricity transfer 185199
- Demystifying non-excellence 193207
- Appendix A. Morley’s omitting types theorem 205219
- Appendix B. Omitting types in uncountable models 211225
- Appendix C. Weak diamonds 217231
- Appendix D. Problems 223237
- Bibliography 227241
- Index 233247 free
- Back Cover Back Cover1251

#### Readership

Graduate students and research mathematicians interested in logic and model theory.