Alfred Tarski, Steven Givant's A Formalization of Set Theory without Variables (Colloquium PDF

By Alfred Tarski, Steven Givant

ISBN-10: 0821810413

ISBN-13: 9780821810415

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Example text

If wto: X, then wf- X. if The implication (ii) is quite elementary; its converse (iii) has a much deeper character. Obviously, (i) implies that every semantical notion defined in terms of to: coincides with the corresponding syntactical notion. This yields, in particular, the following conclusions. , true of every realization of ,c. (v) Two sets W, 0 are equivalent relative to a set iff they are semantically equivalent relative to , i. , every model of which is a model of one of the sets W, 0, is also a model of the other.

It may happen, however, that as a consequence of this replacement some Uk occurs bound in the resulting formula Y at a place where Xk occurs free in X; this is a situation which we want to avoid for rather obvious reasons, in particular, because in this case the connection between substitution and derivability pointed out above may easily fail. If such an undesirable situation occurs, the construction becomes somewhat more complicated. We first select new variables Wo, ... , Wn-l not occurring in X and such that Wk = WI iff Uk = UI, for 0 ::::: k < l < n; to make the selection unambiguous we assume that Wo, ...

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. (vi) For every quantifier-free X E there is aCE II such that X =+ (C = 1). The results (ii)-{vi) originate with Schroder [1895], pp. 150- 153. We now deal briefly with the problem of introducing semantical notions for the language L + .

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

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