Difference between revisions of "Banach-Tarski Paradox"
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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 | + | Note that, since the pieces must be non-measurable, this cannot be done with an actual object. |
==External Links== | ==External Links== | ||
Revision as of 22:28, January 19, 2008
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.
External Links
- Banach-Tarski Paradox at MathWorld