•It implies the Banach-Tarski Paradox (presented by Banach and Tarski in 1924): A ball in3-dimensional space can be decomposed into finitely many pieces (5, actually) in such a waythat those 5 pieces can be reassembled through rigid motions (translation and rotation) intotwo balls of the same size as the original.•If the axiom were false, then other strange things would result. For example, there wouldbe two sets with the property that neither could be put in one-to-one correspondence witha 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 thereis 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 nological contradiction is introduced by assuming thenegationof 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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Mathematical logic, Axiom, Kurt Gödel, Axiom of choice