On unions of chains of models

Abstract

(*) IIxi, • • • , x„Syi, • • • , ym v where v is quantifier-free, if and only if K is closed under unions of chains of models.1 We have adopted most of the notational features of Tarski [7], in which can be found the definitions of relational systems, subsystems, and unions of a chain of systems; we have assumed familiarity with the symbolism of [7]. We have also adopted the convention of giving definitions and theorems only for a similarity class of systems of the type (A, S) where 5 is a ternary relation over A. All the results we have obtained here can be carried over trivially for similarity classes of systems in which there are a finite number of finitary relations. It will be apparent to the reader that this paper exploits a special method and a special theorem. The method is that of graph-diagrams first introduced in Robinson [6], and the theorem is the completeness theorem of Godel [3]. We shall use the completeness theorem in the generalized form (by Malcev and Henkin) where we allow an arbitrary number of constants in the first-order predicate calculus. The plan of the paper is first to give a mathematical characterization of those classes K which are determined by sentences of the form (*), then to use this characterization to prove our main result.2 At the end, we shall apply our result to convex arithmetical classes.