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Download e-book for iPad: A Formalization of Set Theory without Variables (Colloquium by Alfred Tarski, Steven Givant

By Alfred Tarski, Steven Givant

ISBN-10: 0821810413

ISBN-13: 9780821810415

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

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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}! for some finite set I}! From the above definition it is seen that f>77 0 coincides with the set of all logically provable sentences and is the least theory in ,c.

We prefer, however, partly for typographical reasons, to change the symbolism in the opposite direction: we extend the use of "f-" so as to cover all the situations in which we have employed "f-+" until now. This includes, in particular, such contexts as "\II f- X". We also make an analogous stipulation concerning "=== " and " ===+ " . We may, of course, return to the original symbolism in some exceptional cases, so as to enhance the clarity of the text. Notice that 8q\ll cannot be identified with 8q+\II , even if \II E.

4. All the formalisms with which we shall be concerned are assumed to be semantically sound, and are usually easily seen to be so. Not all of them, however, are semantically adequate. 4 do not coincide in general with the corresponding syntactical notions. Instead, a kind of implication or inclusion holds as a rule between any two corresponding notions. e. , they have all models in common. However, the implication in the opposite direction may fail. In the present work, syntactical notions playa much more essential role than the semantical ones.

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

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