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You are not logged in. #1 20080829 20:40:01
Axiom of choiceThis result could not be intuitively simpler, yet it depends on the axiom of choice for its truth! Let be the set of all equivalence classes under the relation ~ on A defined by . Without the axiom of choice, we would not be able to take from each member of one and only one element to form our subset C. (More precisely, it is the axiom of choice that guarantees the existence of an injective function , from which we set .) #2 20080830 09:47:33
Re: Axiom of choiceZorn's lemma is obviously true, the Wellordering principle is obviously false, and who knows about the Axiom of Choice. "In the real world, this would be a problem. But in mathematics, we can just define a place where this problem doesn't exist. So we'll go ahead and do that now..." 