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Anatoly_Vorobey comments on Explained: Gödel's theorem and the Banach-Tarski Paradox - Less Wrong Discussion
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You get real understanding when you study the actual proof. For that, the best book I know is Smullyan's Goedel's Incompleteness Theorems. For an informal argument that can be understood by someone who doesn't know what a formal system is, I think this one is quite good.