Mathematics/Trivia: Difference between revisions

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** This paradox was the subject of an [[wikipedia:Non-measurable set|April Fool's joke in Scientific American magazine]].

** Also seen expressed thuslywise: "What's an anagram of 'Banach-Tarski'? 'Banach-Tarski Banach-Tarski'."

** Another catch is that this stems from a axiom that's commonly ''omitted'' from set theory, being totally consistent with and independent of the others, but frequently yielding paradoxical results like these. You can choose to use it or not use it, similar to the parallel postulate.

** Another catch is that this stems from an axiom that's commonly ''omitted'' from set theory, being totally consistent with and independent of the others, but frequently yielding paradoxical results like these. You can choose to use it or not use it, similar to the parallel postulate.

*** However, that is underselling its use. While the axiom of choice (which is the axiom that is vital in the construction) is used all over the place in analysis, and used in a couple mind-numbingly ''crucial'' places in algebra. Almost all ordinary mathematicians (read: non-FOMers) accept choice as true, and see the paradoxical results as unfortunate things that are true, and see the benefit from the more normal use of the axiom.

*** Or you can accept the fact that mathematics is simply a lot less intuitive than we'd guess at first glance, and that there's nothing unfortunate or wrong about Banach-Tarski's paradox... in fact, in hindsight I would be much more surprised if such things ''were not'' possible, since there's no reason why splitting a sphere in non-measurable sets would need to preserve any intuitive notion of volume. Poor Axiom of Choice, always wronged for no real reason.