Provability with finitely many variables

In first-order logic with equality but with finitely many variables, no finite schema suffices to give a sound and complete axiomatization of the universally valid sentences. The proof uses a rather deep result from algebraic logic. The purpose of this note is to give a rather obvious consequence of a recent theorem of J. S. Johnson [5] (whose proof is based on Monk [7]). The consequence has to do with provability in languages which differ from ordinary first-order languages with equality just in having only finitely many individual variables. Such languages have previously been investigated in Henkin [2 ], Henkin-Tarski [3 ], Jaskowski [4], and Pieczkowski [8]. Our result is that it is impossible to write down finitely many schemata which will give a notion of proof that is sound and complete. See below for a more precise formulation; in particular, the notion of schema is made explicit. What makes our result an easy consequence of Johnson's theorem is the fairly general knowledge of certain connections between logically valid sentences in these restricted first-order languages, and equations which hold identically in each representable polyadic equality algebra. The main portion of the note is devoted to an exposition of these connections. Thanks are due to the referee for aid in the formulation of these connections. We employ the usual set-theoretic notation. f*X is the f-image of the set X. co is the set of all natural numbers. We assume throughout that 3 -<a <& (Our languages will have a variables; the case a <2 is considered in Henkin [2].) ?a is the first-order language with equality with the sequence (vi:i<a) of individual variables and with the sequence (Ri:i<4) of nonlogical constants, where Ri is an a-ary relation symbol for each i<co. We treat 7, -, V, and = as primitive logical symbols; V, A/, +*, and 3 are defined in the usual way. An 5a-structure is a structure 21= (A, Ri)i<,, where A 70 and RiCaA for each i<cw. If Received by the editors November 10, 1969 and, in revised form, March 13, 1970. AMS 1969 subject classifications. Primary 0216, 0218; Secondary 0248.