Kirby J. An Invitation to Model Theory

Cambridge: Cambridge University Press, 2019. — 198 p. — ISBN: 978-1-107-16388-1, 978-1-316-61555-3.This book is designed as an undergraduate or masters level course in model theory. It has grown out of courses taught for many years in Oxford, and courses taught by me at UEA. The choice of material and presentation is based on pedagogical considerations, and I have tried to resist the temptation to be encyclopedic. In this book, the main programme of model theory is to take a familiar mathematical structure and get an understanding of it in the following way. First, find an axiomatisation of its complete theoryLanguages and SoructuresStructures
Terms
Formulas
Definable Sets
Substructures and QuantifiersTheories and CompactnessTheories and Axioms
The Complex and Real Fields
Compactness and New Constants
Axiomatisable Classes
Cardinality Considerations
Constructing Models from SyntaxChanging ModelsElementary Substructures
Elementary Extensions
Vector Spaces and Categoricity
Linear Orders
The Successor StructureCharacterising Definable SetsQuantifier Elimination for DLO
Substructure Completeness
Power Sets and Boolean Algebras
The Algebras of Definable Sets
Real Vector Spaces and Parameters
Semi-algebraic SetsTypesRealising Types
Omitting Types
Countable Categoricity
Large and Small Countable Models
Saturated ModelsAlgebraically Closed FieldsFields and Their Extensions
Algebraic Closures of Fields
Categoricity and Completeness
Definable Sets and Varieties
Hilbert’s Nullstellensatz