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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 is called a paradox because it violates our physical intuition that this is impossible.  It is very similar to the [[Vitali set]], which was developed to illustrate possible flaws in using the [[Axiom of Choice]] (an element of [[Zermelo-Fraenkel]] set theory) in order to prove mathematical claims.  Since the Vitali set is less easily discussed with a layman, the Banach-Tarski paradox is often used to illustrate the "weirdness" of the Axiom of Choice.

The '''Banach-Tarski paradox''' is that statement that a sphere can be cut into 5 non-measurable pieces, which can then be reassembled into two spheres of equal volume; however, this cannot be done with an actual object.  It is called a paradox because it violates our physical intuition that this is impossible.  It is very similar to the [[Vitali set]], which was developed to illustrate possible flaws in using the [[Axiom of Choice]] (an element of [[Zermelo-Fraenkel]] set theory) in order to prove mathematical claims.  Since the Vitali set is less easily discussed with a layman, the Banach-Tarski paradox is often used to illustrate the "weirdness" of the Axiom of Choice.  

 

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

==The Mathematics Behind It==

==The Mathematics Behind It==