Re: Banach–Tarski paradox = Contradiction?
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Well, if you had defined it by saying "every onto mapping must also be 1-1" -- a definition that works fine for equality of finite sets, so why not use that one? -- then the whole thing breaks down. Even numbers and integers are no longer equal, and our "intuition" still works.
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Our intuition doesn't work so well under that definition when we realize that the even numbers and the even numbers are no longer equal either.
PairTheBoard
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