Models of Complete Theories

The semantical concepts, such as satisfaction, truth, and model, form the subject matter of a field known as the theory of models. I am going to discuss today several related recent developments in this field. They all lie in one particular area which is indicated by the title and which will be described more fully in a moment. However, some introductory and side remarks I shall make may also serve to indicate to those unfamiliar with the theory of models at least what some of the other areas of the field are. Perhaps the earliest result in the theory of models, dating from 1915, is the theorem of Löwenheim and Skolem: Any infinite algebraic system has a denumerable subsystem having the same true (elementary) sentences. Before discussing this theorem further, we must define the notions involved in it. By an algebraic system is meant a system 21 = (| 2ï|, R%, Rf, • • • ) formed by a nonempty set |2ï| and finitely or denumerably many relations R$, Rf, • • • among the elements of 2t, each R% having a finite number pw of places. Thus, for example, an ordered group is a system (Gt < , •, e) having a binary relation, a binary operation (which may be regarded as a special kind of ternary relation), and a distinguished element (a special kind of singulary relation). A system S3, having the same similarity type p as 21, is a subsystem of 21 if | S31 C j 211 and each R® is R* restricted to | S3|. The cardinal number of 21 (21) means that of |2l|. The symbols of the elementary language Lp are the sentential connectives Ai V, ~ i —», <-*9 the quantifiers V, 3, the individual variables Vo, Vi, • • • , the equality symbol « , and the pn-ary relation symbols 2?o, 2?i, • • • . A typical formula (of Lp), taking p0 = 2, is