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Banach–Tarski paradox



First stated by Stefan Banach and Alfred Tarski in 1924, the Banach-Tarski paradox or Hausdorff-Banach-Tarski paradox is the famous "doubling the ball" paradox, which states that by using the axiom of choice it is possible to take a solid ball in 3-dimensional space, cut it up into finitely many (non-measurable) pieces and, moving them using only rotations and translations, reassemble the pieces into two balls of the same radius as the original. Banach and Tarski intended this proof to be evidence in favor of rejecting the axiom of choice, but the nature of the proof is such that most mathematicians take it to mean that the axiom of choice merely results in bizarre and unintuitive consequences.



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