This 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.
Pretty much everything that involves abusing the Axiom of Choice ends up trippy and fun.
"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.
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 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...