It is a theorem that a solid ball can be split into finitely many pieces and reassembled into two balls of the same size. The pieces are non-measurable point sets, so it cannot be done physically.
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Hilbert's hotel is a thought experiment about a hotel with infinitely many rooms. Though full, it can still house one, infinitely many, or even infinitely many busloads of new guests.
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Russell's paradox shows that the set of all sets that are not members of themselves cannot exist. It exposed a contradiction in naive set theory and prompted new foundations for mathematics.
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