Lectures on Infinitary Model Theory

Explores connections between infinitary model theory and other branches of mathematical logic, with algebraic applications.

David Marker (Author)

9781107181939, Cambridge University Press

Hardback, published 27 October 2016

192 pages, 180 exercises
23.6 x 16 x 1.8 cm, 0.43 kg

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.

Introduction
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 L?1,?
12. Spectra of Vaught counterexamples
Appendix A. N1-free abelian groups
Appendix B. Admissibility
References
Index.

Subject Areas: Algebraic geometry [PBMW], Algebra [PBF], Set theory [PBCH], Mathematical logic [PBCD]