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 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 has been mathematically proven to be consistent with [[Zermelo-Fraenkel]] set theory. | ||
Revision as of 00:18, January 5, 2010
This article contains disputed material. Please join the discussion, or just fix it!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. Note that, since the pieces must be non-measurable, this cannot be done with an actual object.
Interpretation
Some view the Banach-Tarski paradox as an absurd result, and evidence that the Axiom of Choice is false. However, it can also be viewed as a mathematical affirmation of the biblical story of the loaves and the fishes (as Jesus would not have had to violate the conservation of mass, but merely partition loaves and fishes into non-measurable subsets).
External Links
- Banach-Tarski Paradox at MathWorld
