By Alfred Tarski, Steven Givant

ISBN-10: 0821810413

ISBN-13: 9780821810415

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3. Derivability in ,£; An important step in the formalization of predicate logic is the selection of logical axioms. The set of logical axioms is necessarily infinite, but it is usually described as the set of all particular instances of a few simple axiom schemata. Several axiomatizations of predicate logic which are adequate for our purposes can be found in the literature. 3(i) axioms for L the set described (in a somewhat different symbolism) in Tarski [1965], pp. 67- 68; the set will be denoted here by "A[L]".

In are the distinct elements of I. 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.

Rather, the discussions centered around our present notion of a theory. However, this may not be quite clear to a casual reader since in those papers the term "system" did not have its present meaning, but was rather used as a name for what we are now calling "theory". ) On the other hand, the term "theory" was used in a rather loose and informal way, with a meaning rather close to that of "system" in the present work. 6{v) FORMALISMS AND SYSTEMS 21 However, in this book we shall also concern ourselves with systems in formalisms of predicate logic whose theories are not recursively enumerable.

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

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