Some model theory

Abstract

Abstract Model theory studies the class of models of a given theory. We have already encountered two theorems that tend in this direction: the completeness theorem and the powerful compactness theorem, both of which assert that, under certain conditions, this class is not empty. The central notion in this chapter and for the kind of model theory that we will develop here is the notion of elementary substructure. Intuitively, M is an elementary substructure of N if, obviously, M is a substructure of N and if, for every finite sequences of elements of M and for every property F[s] that is expressible by a first-order formula, it is equivalent to verify that s satisfies F in M or that s satisfies F in N. This notion will be our concern for the first two sections; the important results will be the Lowenheim-Skolem theorems and their corollaries which imply that a countable theory that has an infinite model must have infinite models of every infinite cardinality.