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Lectures on Infinitary Model Theory


Part of Lecture Notes in Logic

  • Date Published: October 2016
  • availability: Available
  • format: Hardback
  • isbn: 9781107181939

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About the Authors
  • Infinitary logic, the logic of languages with infinitely long conjunctions, plays an important role in model theory, recursion theory and descriptive set theory. This book is the first modern introduction to the subject in forty years, and will bring students and researchers in all areas of mathematical logic up to the threshold of modern research. The classical topics of back-and-forth systems, model existence techniques, indiscernibles and end extensions are covered before more modern topics are surveyed. Zilber's categoricity theorem for quasiminimal excellent classes is proved and an application is given to covers of multiplicative groups. Infinitary methods are also used to study uncountable models of counterexamples to Vaught's conjecture, and effective aspects of infinitary model theory are reviewed, including an introduction to Montalbán's recent work on spectra of Vaught counterexamples. Self-contained introductions to effective descriptive set theory and hyperarithmetic theory are provided, as is an appendix on admissible model theory.

    • The first modern introduction to infinitary model theory in forty years, bringing students to the threshold of modern research
    • A comprehensive treatment, including applications to Vaught's conjecture and effective descriptive set theory
    • Introduces Zilber's applications to algebraic geometry
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    Product details

    • Date Published: October 2016
    • format: Hardback
    • isbn: 9781107181939
    • length: 192 pages
    • dimensions: 236 x 160 x 18 mm
    • weight: 0.43kg
    • contains: 180 exercises
    • availability: Available
  • Table of Contents

    Part I. Classical Results in Infinitary Model Theory:
    1. Infinitary languages
    2. Back and forth
    3. The space of countable models
    4. The model existence theorem
    5. Hanf numbers and indiscernibles
    Part II. Building Uncountable Models:
    6. Elementary chains
    7. Vaught counterexamples
    8. Quasinimal excellence
    Part III. Effective Considerations:
    9. Effective descriptive set theory
    10. Hyperarithmetic sets
    11. Effective aspects of Lω1,ω
    12. Spectra of Vaught counterexamples
    Appendix A. N1-free abelian groups
    Appendix B. Admissibility

  • Author

    David Marker, University of Illinois, Chicago
    David Marker is LAS Distinguished Professor of Mathematics at the University of Illinois, Chicago, and a Fellow of the American Mathematical Society. His main area of research is model theory and its connections to algebra, geometry and descriptive set theory. His book, Model Theory: An Introduction, is one of the most frequently used graduate texts in the subject and was awarded the Shoenfield Prize for expository writing by the Association for Symbolic Logic.

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