Banach-Tarski paradox: Difference between revisions
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The Banach-Tarski is a famous, counterintuitive consequence of the [[axiom of choice]]. It says that it's possible to decompose a ball in three-dimensional space into separate parts, which can be rearranged to form two balls, each with the same volume as the original. However, an actual such decomposition in the real world is not possible, since the separate parts aren't actual shapes. |
The Banach-Tarski is a famous, counterintuitive consequence of the [[axiom of choice]]. It says that it's possible to decompose a ball in three-dimensional space into separate parts, which can be rearranged to form two balls, each with the same volume as the original. However, an actual such decomposition in the real world is not possible, since the separate parts aren't actual shapes. |
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Latest revision as of 16:48, 25 March 2024
The Banach-Tarski is a famous, counterintuitive consequence of the axiom of choice. It says that it's possible to decompose a ball in three-dimensional space into separate parts, which can be rearranged to form two balls, each with the same volume as the original. However, an actual such decomposition in the real world is not possible, since the separate parts aren't actual shapes.
The proof requires the axiom of choice, and, therefore, the truth of the Banach-Tarski paradox is a common argument against the usage of the axiom of choice.