•
It implies the BanachTarski Paradox (presented by Banach and Tarski in 1924): A ball in
3dimensional 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 onetoone correspondence with
a subset of the other. That is, there would be two sets of incomparable sizes.
•
In 1940, Gödel 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.
•
Together, Gödel’s and Cohen’s work showed that the Axiom of Choice is independent of ZF:
ZF can’t be used to prove or disprove it.
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 Fall '10
 wen
 Logic, Mathematical logic, Axiom, Kurt Gödel, Axiom of choice

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