An Invitation to Model Theory

Häftad, Engelska, 2019

AvJonathan Kirby

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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.

Produktinformation

Utgivningsdatum2019-04-18
Mått213 x 229 x 10 mm
Vikt272 g
FormatHäftad
SpråkEngelska
Antal sidor194
FörlagCambridge University Press
ISBN9781316615553

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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.

'… this book is well suited for two different audiences: advanced researchers in mathematics who would like to get a first acquaintance with some of the notions and results in model theory without having to spend a considerable amount of time with some classical notions whose relevance may not be clear at the beginning. The second audience is students (or rather, those faculty members considering offering an introductory course in model theory) at universities which do not offer many courses in mathematical logic, but are interested in broadening their curricula with a first introductory course in model theory.' A. Martin-Pizarro, The Bulletin of Symbolic Logic