Mathematics/Trivia: Difference between revisions
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*** 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.
** [[Irregular Webcomic|You'd think they could do it in just four pieces though.]]
* And then there's Whitehead and Russell's ''[[wikipedia:Principia Mathematica|Principia Mathematica]]'', a multi-volume opus intended to construct all of mathematics from the most basic of axioms (such as "every statement is either true or false"). After 379 pages of incomprehensibly dense notation, it succeeds in proving that
** [[Understatement|"The above proposition is occasionally useful"]] was their comment when the 1+1=2 proof was finally complete (presumably after arithmetical addition had been defined) in the second volume.
** It bears pointing out that Gödel's Incompleteness Theorem essentially says that no axiomatic system can be both complete and consistent, ''i.e.'', your system can be complete or consistent but not both. Given that, most mathematicians choose the latter, accepting an incomplete axiom system whose incompleteness is tempered by consistency (so that, ''e.g.'', 1+1 does indeed equal 2).
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