By Alfred Tarski, Steven Givant

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We do not assume, in general, that the finitary part, or even the singleton part, of the derivability relation of a formalism is recursively enumerable (cf. 3, where this terminology is introduced). Whenever this assumption is essential, it will be explicitly stated. However, it is worth pointing out that this assumption does indeed hold for almost all formalisms discussed in this book. In this section we consider an arbitrary system S(1) = (E(l), f-(l), RE(l), MO(1)) and some extension of it, (cf.

Tpk-d and sois 1 , where Gk is a predicate of rank pk > 0 and to, ... , tpk-1, So, Sl are arbitrary terms. Formulas are formed in the familiar way from atomic formulas. 3 for L, remain in P virtually unchanged. 4 can now be carried over from L to P with only minor modifications. Just as L, the formalism P is semantically complete. For the discussion in our work, the assumption that L has just one nonlogical constant is not essential. 6 systems developed in such formalisms . The assumption that all nonlogical constants are binary predicates is essential.

Several axiomatizations of predicate logic which are adequate for our purposes can be found in the literature. 3(i) axioms for L the set described (in a somewhat different symbolism) in Tarski [1965], pp. 67- 68; the set will be denoted here by "A[L]". A precise definition of A[L] follows. (i) A[L], x, Y, Z E OT simply A, is the set of sentences 8 in :E such that, for some 41 and some x, y E 1', 8 coincides with one of the following sentences: (AI) (All) (AIII) (AIV) (AV) (AVI) (AVII) (AVIII) (AIX) [(X - Y) - ((Y - Z) - (X - Z))); [(-,X _ X) - X ); [X - (-,X - Y) ); [VxVyX - VyVxX); [Vx(X - Y) - (VxX - VxY)); [VxX-X); [X - VxX), where x tJ.

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