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,

Related Topics:
Stefan Banach - Alfred Tarski - 1924 - Paradox

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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 for this proof to demonstrate that the axiom of choice was incorrect, 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.

Related Topics:
Axiom of choice - Ball - Finite - Measurable - Rotation - Translation

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