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
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.
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