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The '''Banach-Tarski paradox''' is that statement that a ball can be cut into 5 pieces, which can then be reassembled into two balls of equal volume.  It relies on the [[Axiom of Choice]].  It is called a paradox because it violates our physical intuition that this is impossible, though this paradox has been proven under [[Zermelo-Fraenkel]] set theory.

The '''Banach-Tarski paradox''' is that statement that a ball can be cut into 5 pieces, which can then be reassembled into two balls of equal volume.  It is called a paradox because it violates our physical intuition that this is impossible.  This was developed to demonstrate doubts about using the [[Axiom of Choice]] (an element of [[Zermelo-Fraenkel]] set theory) in order to prove mathematical claims.

Note that, since the pieces must be non-measurable, this cannot be done with an actual object.

Note that, since the pieces must be non-measurable, this cannot be done with an actual object.