An Invitation to Model Theory

Jonathan Kirby (University of East Anglia)

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English

Cambridge University Press
18 April 2019

Philosophy: logic; Mathematics & Sciences; Philosophy of mathematics; Mathematical logic; Set theory; Mathematical modelling

Model theory begins with an audacious idea: to consider statements about mathematical structures as mathematical objects of study in their own right. While inherently important as a tool of mathematical logic, it also enjoys connections to and applications in diverse branches of mathematics, including algebra, number theory and analysis. Despite this, traditional introductions to model theory assume a graduate-level background of the reader. In this innovative textbook, Jonathan Kirby brings model theory to an undergraduate audience. The highlights of basic model theory are illustrated through examples from specific structures familiar from undergraduate mathematics, paying particular attention to definable sets throughout. With numerous exercises of varying difficulty, this is an accessible introduction to model theory and its place in mathematics.

By: Jonathan Kirby (University of East Anglia)
Imprint: Cambridge University Press
Country of Publication: United Kingdom
Dimensions: Height: 227mm, Width: 151mm, Spine: 12mm
Weight: 300g
ISBN: 9781316615553
ISBN 10: 1316615553
Pages: 195
Publication Date: 18 April 2019
Audience: College/higher education , Professional and scholarly , A / AS level , Undergraduate
Format: Paperback
Publisher's Status: Active

Preface; Part I. Languages and Structures: 1. Structures; 2. Terms; 3. Formulas; 4. Definable sets; 5. Substructures and quantifiers; Part II. Theories and Compactness: 6. Theories and axioms; 7. The complex and real fields; 8. Compactness and new constants; 9. Axiomatisable classes; 10. Cardinality considerations; 11. Constructing models from syntax; Part III. Changing Models: 12. Elementary substructures; 13. Elementary extensions; 14. Vector spaces and categoricity; 15. Linear orders; 16. The successor structure; Part IV. Characterising Definable Sets: 17. Quantifier elimination for DLO; 18. Substructure completeness; 19. Power sets and Boolean algebras; 20. The algebras of definable sets; 21. Real vector spaces and parameters; 22. Semi-algebraic sets; Part V. Types: 23. Realising types; 24. Omitting types; 25. Countable categoricity; 26. Large and small countable models; 27. Saturated models; Part VI. Algebraically Closed Fields: 28. Fields and their extensions; 29. Algebraic closures of fields; 30. Categoricity and completeness; 31. Definable sets and varieties; 32. Hilbert's Nullstellensatz; Bibliography; Index.

Jonathan Kirby is a Senior Lecturer in Mathematics at the University of East Anglia. A student of Boris Zilber, his main research is in model theory and its interactions with algebra, number theory, and analysis, with particular interest in exponential functions. He has taught model theory at Oxford, the University of Illinois at Chicago, and the University of East Anglia.