history-page47 - It implies the Banach-Tarski Paradox...

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Unformatted text preview: It implies the Banach-Tarski Paradox (presented by Banach and Tarski in 1924): A ball in 3-dimensional space can be decomposed into finitely many pieces (5, actually) in such a way that those 5 pieces can be reassembled through rigid motions (translation and rotation) into two balls of the same size as the original. If the axiom were false, then other strange things would result. For example, there would be two sets with the property that neither could be put in one-to-one correspondence with a subset of the other. That is, there would be two sets of incomparable sizes. In 1940, Gdel proved that if there is no logical contradiction in the axioms of ZF then there is no logical contradiction in the axioms of ZFC. In 1963, Cohen proved that if there is no logical contradiction in the axioms of ZF then no logical contradiction is introduced by assuming the negation of the Axiom of Choice....
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This note was uploaded on 12/29/2011 for the course MATH 378 taught by Professor Wen during the Fall '10 term at SUNY Stony Brook.

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