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This might be too broad question - but what would be the things that you consider as surpising facts and discoveries in various areas of mathematics?

If this seems too broad, it is fine to stick with probability theory.

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@MarcvanLeeuwen In. – Maps from subways Mar 27 '13 at 10:44
@MarcvanLeeuwen I think a fact in, e.g., probability theory is also a fact about probability theory. – Quinn Culver Mar 27 '13 at 13:15
@QuinnCulver: It's corrected now, never mind. – Marc van Leeuwen Mar 27 '13 at 13:17

I would like you to go to this link to a similar post on quora.

Perhaps, the most impressive mathematical discovery ever made is Gödel's incompleteness theorems, ostensibly because they demonstrate that a complete and consistent set of axioms for a given branch of mathematics is impossible. This doesn't in anyway diminish the power of mathematics, it simply states that there are limits to any sub-discipline of maths.

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