A Formalization of Set Theory without Variables (Colloquium - download pdf or read online

By Alfred Tarski, Steven Givant

ISBN-10: 0821810413

ISBN-13: 9780821810415

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More generally, given a class K of such structures ll, the theory of K, 0pK, is the intersection of all 0pll for II E K. A sentence X E E is called a consequence of a set \[I <;::; E , and \[I is said to imply X, in symbols \[I F X , if every model of \[I is a model of X . 3 in terms of derivability. In opposition to the new, semantical notions, those previously defined are referred to as syntactical. 4(vii) £, 13 part of the original term, or else we introduce a new term, without following the general rule.

Iii) Finally, we shall simply say that S is a system if there is a formalism :1' such that S is a system developed in :1'. As a direct consequence of (i)-(iii) we obtain (iv) Every formalism :1' is a system, and in fact the system in :1' with base 0. (Since, by definition, MOX[31 is a subclass of RE[31, we are assuming that n{MOX[31: X E 0} = RE[31; cf. ) By (iv) every result established for all systems holds a fortiori for all formalisms. Various notions and notations introduced for the formalism L, and carried over in the first part of this section to arbitrary formalisms :1', can be extended in an obvious way to arbitrary systems S.

Un-l for xo, . ,Xn-l, and the result of performing this operation on X is denoted by "X[xo/uo, ... , xn-l/un-d". 3(ii). In fact, the notion of substitution is defined in such a way that, for every formula X, the expression X[xo/uo, ... , xn-l/un-d is a formula whose closure is derivable from the closure of X. Throughout the present work, however, the notion of substitution will not be extensively used, but will prove very helpful in some fragments of our discussion. The substitution X[xo/uo, ...

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

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