Banach-Tarski paradox: http://en.wikipedia.org/wiki/Banach–Tarski_paradoxThis has always blown my mind. From Wikipedia: "The Banach–Tarski paradox is a theorem in set theoretic geometry which states that a solid ball in 3-dimensional space can be split into several non-overlapping pieces, which can then be put back together in a different way to yield two identical copies of the original ball."Trippy.

 Leading to one of my co-worker's favourite jokes, Q: What's an anagram of "Banach-Tarski" A: "Banach-Tarski Banach-Tarski"
 You can also deconstruct and rebuild that ball into another one of infinite size, I believe. I'm not sure if you can do infinite off the top of my head, but I'm certain of at least arbitrary.Pretty much everything that involves abusing the Axiom of Choice ends up trippy and fun.
 My favorite quote about the Axiom of Choice[1]:"The Axiom of Choice is obviously true, the Well-Ordering Principle is obviously false, and who can tell about Zorn's Lemma?"Of course, AOC, WOP, and ZL are all logically equivalent.[1] Possibly the nerdiest thought my brain has ever formed; this site makes me very happy.
 I'd really like someone to explain Banach-Tarski to me in some way that I can sort of understand. I understand it has to do with AOC, which seems totally reasonable, and then I'm lost.Also, I have trouble with "incrementally" understanding this. I can understand that the limit of 1/n (n->INF) is 0, because even with large finite numbers we're getting close to 0, but no matter how many times I cut up an orange I detect no Banach-Tarski strangeness.
 I honestly have a difficult time understanding the AC paradoxes, too. I recognize that it's totally necessary for many things (i.e. multidimensional integrals) and when it's consistent with a logical conclusion I have little trouble "seeing" it. Otherwise, though, not so much.I think it's interesting to see the connection between the AC and the Law of Excluded Middle which also causes paradoxical issues from time to time.http://en.wikipedia.org/wiki/Axiom_of_choice#Law_of_the_excl...

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