Wolfram stating Gödel's second incompleteness theorem informally:Any formal system that is interesting enough to formulate its own consistency can prove its own consistency iff it is inconsistent.The iff isn't a misspelling, it means "if and only if" for those who may not know.
I love this theorem, a source of such great philosophizing. :)
Curious, I consistently attempt to be interesting, perhaps I should be more inconsistent. Of course I am not a mathematical axiom so this can't apply to me; it is inconsistent.
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I like what von Neumann wrote: "Thus, I think that your result has solved negatively the foundational question: there is no rigorous justification for classical mathematics."