Difference between revisions of "Banach-Tarski Paradox"
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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.) |
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| − | The '''Banach-Tarski paradox''' uses the [[Axiom of Choice]] to take a [[sphere]], split it into | + | 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. | ||
Revision as of 23:15, January 4, 2010
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.
External Links
- Banach-Tarski Paradox at MathWorld
