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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.

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.

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.

*[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:Mathematics]]

[[Category:Set Theory]]

[[Category:geometry]]