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The Banach-Tarski paradox uses the [[Axiom_of_Choice|Axiom of Choice]] to take a sphere, split it into finitely many pieces, and reassemble them into two new spheres. The new spheres both have the same volume as the original, suggesting paradoxically that you can create two identical copies out of one, despite our physical intuition that this is impossible. This paradox is has been mathematically proven to be consistent with [[Zermelo-Fraenkel|Zermelo-Fraenkel]] set theory.

The Banach-Tarski paradox uses the [[Axiom of Choice]] to take a sphere, split it into finitely many pieces, and reassemble them into two new spheres. The new spheres both have the same volume as the original, suggesting paradoxically that you can create two identical copies out of one, despite our physical intuition that this is impossible. This paradox is has been mathematically proven to be consistent with [[Zermelo-Fraenkel]] set theory.

Of course, we need only consider the Biblical story of the loaves and fishes to realize that the Banach-Tarski paradox is consistent. Jesus Himself showed that such a feat is possible (but miraculously so):

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

''He replied, "You give them something to eat." They answered, "We have only five loaves of bread and two fish—unless we go and buy food for all this crowd." (About five thousand men were there.) But he said to his disciples, "Have them sit down in groups of about fifty each." The disciples did so, and everybody sat down. Taking the five loaves and the two fish and looking up to heaven, he gave thanks and broke them. Then he gave them to the disciples to set before the people. They all ate and were satisfied, and the disciples picked up twelve basketfuls of broken pieces that were left over.'' Luke 9:13-17 ([[http://www.biblegateway.com/passage/?search=Lk%209:13-17&version=31; New International Version]])

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

[[Category:Mathematics]]

[[Category:Mathematics]]