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

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