By Alfred Tarski, Steven Givant
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Additional resources for A Formalization of Set Theory without Variables (Colloquium Publications)
Corollary (v) suggests a natural way of extending the relation of equivalence under a set Ill- and, in particular, that of logical equivalence- from sentences to arbitrary formulas. In fact, we can stipulate that , for any formulas X , Y E and any set of sentences III X ==w Y iff III f- [X ++ Y), and, in particular, X == Y iff f- [X ++ Y) . We do not extend the relations ==w and == to arbitrary sets of formulas. " and to relations between and operations on these expressions. As a consequence, such sets as 1', prove to be recursive.
Formal languages would then be construed as structures with a different list of fundamental components; the list would include some notions referring to the intrinsic structure of sentences, such as the vocabulary of a language. There is another notion of a general character, closely related to the notion of a formalism, that will frequently be used in this work, namely the notion of a system. Just as in the case of formalisms, we are not planning to construct a general theory of systems. Actually, for the purposes of this work, we restrict ourselves to those systems which are developed in a formalism or, what amounts to the same thing, to systems obtained by relativizing a formalism to a certain set of sentences.
The set of all predicates is denoted by "n [£ + 1" 23 or simply by "n" . 1(ii) In analogy with the definitions of a formula and of derivability, the above definition of a predicate carries with it a method of showing that every predicate in L + possesses a certain property; this method of proof may be referred to as proof by induction on predicates (cf. 3(ii)). By the definition of predicates just stated, we have A + B , A - , A <:> B , E II whenever A, BEn . Thus +, 0 can be treated as binary operations, and -, as unary operations, on and to predicates.
A Formalization of Set Theory without Variables (Colloquium Publications) by Alfred Tarski, Steven Givant