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Read e-book online A Formalization of Set Theory without Variables (Colloquium PDF

By Alfred Tarski, Steven Givant

ISBN-10: 0821810413

ISBN-13: 9780821810415

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Additional resources for A Formalization of Set Theory without Variables (Colloquium Publications)

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F- X for every X E O. We write "f- X" (or " f- 0") instead of "0 f- X" (or "0 f- 0") and we say in this case that X is a logically provable sentence (or 0 is a set of logically provable sentences) . Given a I}! E , we set f)771}! ['c]) = {X: X E E and I}! f- X}. f>771}! is referred to as the theory generated by, or based upon I}!. A set 0 is called E or, equivalently, if 0 = f>770. Every set a theory if 0 = f)771}! for some I}! I}! E such that 0 = f)771}! is said to be a base, or a (possible) axiom set of 0; 0 is called finitely based or finitely axiomatizable if 0 = f)771}!

And <:> , has priority over each of the two additions , + and e ; (3) in case neither (1) nor (2) is applicable, the operations should be performed successively from left to right. Each of the binary operations mentioned above extends by the usual recursion to an operation on finite sequences of predicates. Thus, for every sequence (A o, ... , An-I) of predicates we set The operations . and Ao+ '" + A n - l = 0 if n = 0, =Ao if n = 1, = (Ao+ · · · + A n - 2 )+A n - l if n = 2,3, .. . By changing, respectively, in these formulas" + " to ".

1{i) in mind, we first define by a recursion on predicates what is meant by saying that a given relation R U x U is denoted (in ll) by a given A E II. In case A is an atomic predicate, we stipulate that the relation denoted by A is E if A = E, and is U1Id if A = 1. Then, in full agreement with the content of Axioms (DI)-{DIV), we make four recursive stipulations, one for each of the four operators occurring in L +. Thus, for example, we stipulate that, in case A = B + C (B, CEIl), the relation denoted by A is the union R U S of the relations Rand S respectively denoted by Band C.

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

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