Difference between revisions of "Banach-Tarski Paradox"
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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. | ||
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| + | ==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== | ==External Links== | ||
Revision as of 23:21, January 4, 2010
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
