A Formalization of Set Theory without Variables (Colloquium by Alfred Tarski, Steven Givant

By Alfred Tarski, Steven Givant

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Tarski-Mostowski- Robinson [1953], pp. 15-17, where (vi) and (viii) are stated and proved; (vii) is an obvious corollary of (vi). , a specific set of nonlogical axioms) for S. Using the set <1>, we relativize to S various notions defined above. Thus we call a sentence X E E provable in S if <1> f- X. We say that X is derivable in S from a set III E if III f-<1> X. Two sentences X, Y E E are said to be equivalent in S if X =<1> Y. f>r]<1> is referred to as the theory of S; more generally, we can call 8 a theory in S if 8 is a theory in Land <1> 8.

We correlate with S an extended system S+ which is formalized in £,+ and has again the base cI>. If, in particular, S is an axiomatic system in £, with the axiom set Ae (cf. , are replaced by ===t, f-t, and f- respectively. In this sense we say that systems Sand S+ are equipollent in means of expression and proof. The second equipollence theorem suggests the possibility of some simplification in our symbolism. , whenever \II E and X E E. Hence we could refrain from using the symbol "f-" altogether and replace it everywhere by "f-+ ".

This will be the task of the remaining part of the present chapter. 6 (although most of our observations will apply to uninterpreted formalisms as well). 6 and are semantically sound. We assume that the notion of recursiveness and related notions such as recursive enumerability have been appropriately extended to sets of sentences, and to relations between and operations on these sentences. In particular, we assume that in each formalism the set of all sentences is recursive. 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.

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