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i'm not a mathematician either, but i think the mistake you're making is in neglecting the difference between "subset A of the sphere has as many points as the entire sphere" and "here's a way to cut a sphere into subsets, and reassemble the subsets into two spheres". intuitively, your mapping scheme involves stretching, shearing and discontinuous transforms on the subsets before you get two spheres out of it - it is pretty obvious from the mathematics of infinite sets that you can always do this, so there's no real "paradox" involved.

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