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Impossible? by Julian Havil

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Chapter 18

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THE BANACH–TARSKI PARADOX

God exists since mathematics is consistent, and the Devil exists since we cannot prove it.

Andre Weil

The subject matter of this last chapter simply has to rank as the most counterintuitive result in mathematics and is a fitting finalé to a book devoted to mathematical surprise.

Stefan Banach and Alfred Tarski brought to the world an improvement on a paradox devised by the great topologist Felix Hausdorff, the formalized form of which is often replaced by something fanciful such as:

A solid sphere can be dissected into five pieces and the pieces reassembled to form two complete spheres of exactly the same size ...

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