Properties of Model Classes

Abstract

In this chapter we wish to study properties of model classes. By a model class we mean the class of all models of an axiom system Σ.

First we shall furnish such a class with a topology whose compactness is exactly the content of the Finiteness Theorem 1.5.6. After that, we shall introduce several properties of the model class of an axiom system Σ, whose study can lead, among other things, to the proof of the completeness (recall §1.6) of Σ. We shall carry this out explicitly for a series of theories axiomatized in (1.6).

The properties of Σ or of its model class to be introduced are: categoricity in a fixed cardinality, model completeness and quantifier elimination. The study of these properties is based not only on the possible applicability to the proof of the completeness of a theory, but is, rather, also justified by its usefulness in concrete, mathematical (in particular, algebraic) theories. In this chapter we shall investigate such properties only for the theory of algebraically closed fields; in Chapter 4, other theories will follow.