Countable models and beyond

Model theory is the study of arbitrary L-structures for first-order languages L. It is a sort of generalised algebraic theory of algebraic structures. We talk of a structure M being a model of a set of L-sentences ∑ when M ╞ ∑, and this gives the name to the theory. Actually, in practice, model theory tends to be much more about the structures themselves and the subsets and functions that are definable in those structures by first-order formulas, and much less about first-order sentences, but the term ‘model theory’ seems to be fixed now. This chapter attempts to give a flavour of model theory and presents some of the first theorems. It also contains a considerable amount of preliminary material on countable sets and cardinalities that we have somehow managed to put off until now.

Model theory starts off with the Compactness Theorem for first-order logic, which is phrased entirely in terms of the notion of semantics, ╞, but was proved in the last chapter by an excursion into the realm of formal proofs. The key result guaranteeing the existence of models of a set of sentences ∑ is the Completeness Theorem for first-order logic which provides us with a model M of ∑, under the assumption that ∑ _⊥. We will start by looking at the Completeness Theorem in a little more detail to see what extra it can say for us.