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Correcting a typo; Banach-Tarski requires splitting the sphere into infinitely many pieces. Axiom of choice is not necessary to select elements from a finite collection of sets.
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 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 infinitely 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 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.