Pick some formal system X which can handle basic arithmetic. The basic idea is to take the classic paradox, "This sentence is false," and embed it into maths. T

Pick some formal system X which can handle basic arithmetic. The basic idea is to take the classic paradox, "This sentence is false," and embed it into maths. The sentence actually constructed is something like, "This theorem is not provable within formal system X," and it's phrased as a theorem of formal system X. The main ingredient we need here is to realize that numbers can represent sentences in formal systems. This should be obvious to computer scientists of today, but it was ground-breaking at the time. Another thing we'll need is diagonalization. Some folks get hung up on this one; ultimately, you'll have to convince yourself that it's a valid technique. Hofstadter's explanations, both the long gentle one in GEB and the quick fuzzy one in "I Am A Strange Loop", are much more detailed and correct on this, so I'd advise going back and making sure you understand each step. Also, something that doesn't get enough airtime is Tarski's Undefinability, which says that a formal system afflicted by incompleteness is not capable of talking about truth in certain basic and important ways which would enable a formal system to fully represent itself.