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The '''Banach-Tarski paradox''' is that statement that a sphere can be cut into 5 [[Non-measurable set|non-measurable sets of points]], which can then be reassembled into two spheres 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 set|non-measurable sets of points]], which can then be reassembled into two spheres of equal volume.  It is called a paradox because it violates our physical intuition that this is impossible; however, it should be noted that the "cuts" necessary are infinitely small and precise, and thus cannot be performed by man.  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.  

This paradox -- or insight -- bears a striking resemblance to the multiplication of the loaves, which is the only miracle described in all four [[Gospels]].

This paradox -- or insight -- bears a striking resemblance to the multiplication of the loaves, which is the only miracle described in all four [[Gospels]].