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Axiomatic set theoryâ€™s Axiom of Choice states that for any collection of non-empty sets there exists at least one set containing exactly one element from each of them.
This entails the Banach-Tarski â€œparadoxâ€ wherein a ball, â€™s or any otherâ€™s, can be disassembled and rearranged into two balls equal to the original.
Crucial is the ballâ€™s infinite divisibility. Just as the set of whole numbers is equinumerous to the evens and the odds, so may the infinite set of points in the ball be the same size as the infinite sets of points in each of the newly spawned balls.