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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; 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.  

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]].

==The Mathematics Behind It==

==The Mathematics Behind It==

Some view the Banach-Tarski paradox as an absurd result, and evidence that the [[Axiom of Choice]] is false.  This is the reaction of many non-mathematicians upon hearing this result.  However, as one grows in mathematical understanding one finds it is impossible to avoid - mathematicians have learned to accept the existence of [[non-measurable set]]s, even though these do not match solid objects in the real world.

Some view the Banach-Tarski paradox as an absurd result, and evidence that the [[Axiom of Choice]] is false.  This is the reaction of many non-mathematicians upon hearing this result.  However, as one grows in mathematical understanding one finds it is impossible to avoid - mathematicians have learned to accept the existence of [[non-measurable set]]s, even though these do not match solid objects in the real world.

==External Links==

==External links==

*[http://mathworld.wolfram.com/Banach-TarskiParadox.html Banach-Tarski Paradox] at MathWorld

*[http://mathworld.wolfram.com/Banach-TarskiParadox.html Banach-Tarski Paradox] at MathWorld

[[Category:Set Theory]]

[[Category:Set Theory]]

[[Category:geometry]]

[[Category:Geometry]]