Models and Ultraproducts An Introduction

In this text for first-year graduate students, the authors provide an elementary exposition of some of the basic concepts of model theory--focusing particularly on the ultraproduct construction and the areas in which it is most useful. The book, which assumes only that its readers are acquainted with the rudiments of set theory, starts by developing the notions of Boolean algebra, propositional calculus, and predicate calculus.
Model theory proper begins in the fourth chapter, followed by an introduction to ultraproduct construction, which includes a detailed look at its theoretic properties. An overview of elementary equivalence provides algebraic descriptions of the elementary classes. Discussions of completeness follow, along with surveys of the work of Jónsson and of Morley and Vaught on homogeneous universal models, and the results of Keisler in connection with the notion of a saturated structure. Additional topics include classical results of Gödel and Skolem, and extensions of classical first-order logic in terms of generalized quantifiers and infinitary languages. Numerous exercises appear throughout the text.

Introduction
0. Prerequisites
1. Boolean Algebras
2. Propositional Calculus
3. Predicate Calculus
4. Beginning Model Theory
5. Introducing Ultraproducts
6. Set Theoretic Properties of Ultraproducts
7. Elementary Classes
8. Ultralimits
9. Completeness and Model Completeness
10. Homogeneous and Universal Structures
11. Saturated Models
12. Applications of Ultraproducts
13. Generalized Quantifiers
14. Infinitary Languages
Some Suggestions for Further Reading
Bibliography
Index of Named Theorems
General Index