Explaining Gödel's Incompleteness Theorem to a Twelve Year Old
i'm not a mathematician, but his final point about the continuum hypothesis seemed odd to me.  the CH is "outside" ZF, but that just means (afaik) that it contains some extra "information" that is not in ZF.  adding it to ZF doesn't force any kind of contradiction, in the way that adding "this system is consistent" would.  so CH is (just) an example of an independent axiom - it doesn't illuminate what is so weird about the completeness theorem.(mathematicians: is that right?)