Item Successfully Added to Cart
An error was encountered while trying to add the item to the cart. Please try again.
OK
Please make all selections above before adding to cart
OK
Share this page via the icons above, or by copying the link below:
Copy To Clipboard
Successfully Copied!
Categoricity
 
John T. Baldwin University of Illinois at Chicago, Chicago, IL
Front Cover for Categoricity
SoftcoverISBN:  978-0-8218-4893-7
Product Code:  ULECT/50
List Price: $62.00
MAA Member Price: $55.80
AMS Member Price: $49.60
eBookISBN:  978-1-4704-1645-4
Product Code:  ULECT/50.E
List Price: $58.00
MAA Member Price: $52.20
AMS Member Price: $46.40
SoftcoverISBN:  978-0-8218-4893-7
eBookISBN:  978-1-4704-1645-4
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
Front Cover for Categoricity
Click above image for expanded view
  • Front Cover for Categoricity
  • Back Cover for Categoricity
Categoricity
John T. Baldwin University of Illinois at Chicago, Chicago, IL
Softcover ISBN:  978-0-8218-4893-7
Product Code:  ULECT/50
List Price: $62.00
MAA Member Price: $55.80
AMS Member Price: $49.60
eBook ISBN:  978-1-4704-1645-4
Product Code:  ULECT/50.E
List Price: $58.00
MAA Member Price: $52.20
AMS Member Price: $46.40
Softcover ISBN:  978-0-8218-4893-7
eBookISBN:  978-1-4704-1645-4
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 Details
     
     
    University Lecture Series
    Volume: 502009; 235 pp
    MSC: Primary 03;

    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.

    Readership

    Graduate 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 non-excellence
    • Appendix A. Morley’s omitting types theorem
    • Appendix B. Omitting types in uncountable models
    • Appendix C. Weak diamonds
    • Appendix D. Problems
  • Requests
     
     
    Review Copy – for reviewers who would like to review an AMS book
    Permission – for use of book, eBook, or Journal content
    Accessibility – to request an alternate format of an AMS title
Volume: 502009; 235 pp
MSC: Primary 03;

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.

Readership

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 non-excellence
  • Appendix A. Morley’s omitting types theorem
  • Appendix B. Omitting types in uncountable models
  • Appendix C. Weak diamonds
  • Appendix D. Problems
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Please select which format for which you are requesting permissions.