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Softcover ISBN:  9780821848937 
Product Code:  ULECT/50 
List Price:  $62.00 
MAA Member Price:  $55.80 
AMS Member Price:  $49.60 
eBook ISBN:  9781470416454 
Product Code:  ULECT/50.E 
List Price:  $58.00 
MAA Member Price:  $52.20 
AMS Member Price:  $46.40 
Softcover ISBN:  9780821848937 
eBookISBN:  9781470416454 
Product Code:  ULECT/50.B 
List Price:  $120.00$91.00 
MAA Member Price:  $108.00$81.90 
AMS Member Price:  $96.00$72.80 

Book DetailsUniversity Lecture SeriesVolume: 50; 2009; 235 ppMSC: Primary 03;
Modern model theory began with Morley's categoricity theorem: A countable firstorder 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 firstorder, 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, omittingtypes theorems, multidimensional 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.ReadershipGraduate students and research mathematicians interested in logic and model theory.

Table of Contents

Part 1. Quasiminimal excellence and complex exponentiation

Chapter 1. Combinatorial geometries and infinitary logics

Chapter 2. Abstract quasiminimality

Chapter 3. Covers of the multiplicative group of $\mathbb {C}$

Part 2. Abstract elementary classes

Chapter 4. Abstract elementary classes

Chapter 5. Two basic results about $L_{\omega _1,\omega }(Q)$

Chapter 6. Categoricity implies completeness

Chapter 7. A model in $\aleph _2$

Part 3. Abstract elementary classes with arbitrarily large models

Chapter 8. Galois types, saturation, and stability

Chapter 9. Brimful models

Chapter 10. Special, limit and saturated models

Chapter 11. Locality and tameness

Chapter 12. Splitting and minimality

Chapter 13. Upward categoricity transfer

Chapter 14. Omitting types and downward categoricity

Chapter 15. Unions of saturated models

Chapter 16. Life without amalgamation

Chapter 17. Amalgamation and few models

Part 4. Categoricity in $L_{\omega _1,\omega }$

Chapter 18. Atomic AEC

Chapter 19. Independence in $\omega $stable classes

Chapter 20. Good systems

Chapter 21. Excellence goes up

Chapter 22. Very few models implies excellence

Chapter 23. Very few models implies amalgamation over pairs

Chapter 24. Excellence and *excellence

Chapter 25. Quasiminimal sets and categoricity transfer

Chapter 26. Demystifying nonexcellence

Appendix A. Morley’s omitting types theorem

Appendix B. Omitting types in uncountable models

Appendix C. Weak diamonds

Appendix D. Problems


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Modern model theory began with Morley's categoricity theorem: A countable firstorder 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 firstorder, 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, omittingtypes theorems, multidimensional 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.
Graduate students and research mathematicians interested in logic and model theory.

Part 1. Quasiminimal excellence and complex exponentiation

Chapter 1. Combinatorial geometries and infinitary logics

Chapter 2. Abstract quasiminimality

Chapter 3. Covers of the multiplicative group of $\mathbb {C}$

Part 2. Abstract elementary classes

Chapter 4. Abstract elementary classes

Chapter 5. Two basic results about $L_{\omega _1,\omega }(Q)$

Chapter 6. Categoricity implies completeness

Chapter 7. A model in $\aleph _2$

Part 3. Abstract elementary classes with arbitrarily large models

Chapter 8. Galois types, saturation, and stability

Chapter 9. Brimful models

Chapter 10. Special, limit and saturated models

Chapter 11. Locality and tameness

Chapter 12. Splitting and minimality

Chapter 13. Upward categoricity transfer

Chapter 14. Omitting types and downward categoricity

Chapter 15. Unions of saturated models

Chapter 16. Life without amalgamation

Chapter 17. Amalgamation and few models

Part 4. Categoricity in $L_{\omega _1,\omega }$

Chapter 18. Atomic AEC

Chapter 19. Independence in $\omega $stable classes

Chapter 20. Good systems

Chapter 21. Excellence goes up

Chapter 22. Very few models implies excellence

Chapter 23. Very few models implies amalgamation over pairs

Chapter 24. Excellence and *excellence

Chapter 25. Quasiminimal sets and categoricity transfer

Chapter 26. Demystifying nonexcellence

Appendix A. Morley’s omitting types theorem

Appendix B. Omitting types in uncountable models

Appendix C. Weak diamonds

Appendix D. Problems