Basic Model Theory

Model theory investigates the relationships between mathematical structures (“models”) on the one hand and formal languages (in which statements about these structures can be formulated) on the other.

Example structures are: the natural numbers with the usual arithmetical operations, the structures familiar from algebra, ordered sets, etc.

The emphasis is on first-order languages, the model theory of which is best known. An example result is Löwenheim's theorem (the oldest in the field): a first-order sentence true of some uncountable structure must hold in some countable structure as well. Second-order languages and several of its fragments are dealt with as well.

As the title indicates, this book introduces the reader to what is basic in model theory. A special feature is its use of the Ehrenfeucht game by which the reader is familiarized with the world of models.

Kees Doets is a lecturer in the department of mathematics, computer science, physics and astronomy at the University of Amsterdam.

Introduction
1 Basic Notes
2 Relations Between Models
- 2.1 Isomorphism and Equivalence
- 2.2 (Elementary) Submodels
3 Ehrenfeucht-Fraïssé Games
- 3.1 Finite Games
- 3.2 The Meaning of the Game
- 3.3 Applications
- 3.4 The Infinite Game
4 Constructing Models
- 4.1 Compactness
- 4.2 Diagrams
- 4.3 Ultraproducts
- 4.4 Omitting Types
- 4.5 Saturation
- 4.6 Recursive Saturation
- 4.7 Applications
A Deduction and Completeness
- A.1 Rules of Natural Deduction
- A.2 Soundness
- A.3 Completeness
B Set Theory
- B.1 Axioms
- B.2 Notations
- B.3 Orderings
- B.4 Ordinals
- B.5 Cardinals
- B.6 Axiom of Choice
- B.7 Inductive Definitions
- B.8 Ramsey's Theorem
- B.9 Games
Bibliography
Name Index
Subject Index
Notation

7/1/96

Basic Model Theory

Contents