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

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193) . We can then define satisfaction for finite sequences by stipulating that x = (xo , . . , Xn-l) satisfies X just in case there is an infinite sequence Y satisfying X such that Xi = Yk i for i = 0, . , n-l. (In the treatment of satisfaction for finite sequences we deviate slightly from the terminology of Henkin- Monk- Tarski , p. ) In terms of satisfaction other semantical notions are defined. The most important among them are the closely related notions of truth and model. A sentence Y E E is said to be true of II or to hold in II if every sequence x = (xo , ...

Similarly, we extend the notion of subformalism. 6(v) (v) A system S is a subsystem of a system 'J, and'J an extension of S, if <;;; and III f- X [S] implies III f- X ['J] whenever III <;;; and X E Notice that the notion of system is used here in the sense of (iii); thus the systems Sand 'J in (v) need not be developed in the same formalism. It may also be pointed out that, for two formalisms, :f and 9, :f is a subformalism of 9 just in case it is a subsystem of 9. c, the set

The theory 81]+0, the logic of L+, may be called the extended predicate logic of one binary relation. We shall state a few interesting logical equivalences which hold between quantifier-free sentences of L+ (where A, B are assumed to be arbitrary predicates in II). (i) (A =B) (ii) (A = 1 - B =1) =+ (A·B+A- ·B- =1). =+ (10A- 01+B = 1). (iii) (-,A=l) =+ (10A-01=1). (iv) (A = 1 V B = 1) =+ (AeOeB = 1). (v) (A=lAB=l) =+ (A·B=l). The proof of (i)-(v) is quite elementary. From (i)- (iii) we easily derive by induction on formulas the following important theorem.