Get A Formalization of Set Theory without Variables (Colloquium PDF

By Alfred Tarski, Steven Givant

ISBN-10: 0821810413

ISBN-13: 9780821810415

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By Alfred Tarski, Steven Givant

ISBN-10: 0821810413

ISBN-13: 9780821810415

Show description

Read or Download A Formalization of Set Theory without Variables (Colloquium Publications) PDF

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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

If Gi is a predicate of rank pi > 0 and xo, ... ,Xpi-1 are arbitrary variables, then the expression formed by the string of symbols Gi , Xo, ... , Xpi-1 and represented here by Gi(XO, ... , xpi-d is an atomic formula of P. In case pi = 2 we can use, as we have done already, xOGiX1 instead of Gi(xo, xd; a similar remark applies to the logical binary predicate 1. In case some of the symbols Gj are operation symbols, the definition of atomic formula undergoes some complications. We first define by recursion the notion of a term: variables and operation symbols of rank 0 are the atomic terms; if Gj is an operation symbol of rank pj > 0 and to, ...

Sets 0 t; :E that satisfy condition (ii)h) above are said to be closed under modus ponens (the operation of detachment). Instead of "{Y} I- X" we write "Y I- X" (and we proceed similarly in other analogous situations). The definition of derivability just formulated suggests a general method of showing that every sentence derivable from a given set IJ! 3(iii) 9 a certain property: we take 0 to be the set of sentences possessing this property, and show successively that 0 satisfies conditions (a)- h) in (ii).

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


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