Comments by likebox I am quite familiar with the completeness theorem It does not show that recursive functions are representable in arithmetic that proof of representabilily is the first third of the proof of the incompleteness theorem There is simply no way to prove the incompletness theorem without some sort of arithmetization result And Godel s original proof did explicitly develop a formula for Bew The idea that the recursion theorem can be avoided by a reference to some sort of quine lemma is interesting but there is no result well known as the quine lemma and the result needed here is not just the existence of a quine but the existence of a fixed point for a certain recursive operator Only the recursion theorem is going to give you that Statements such as It only requires a great deal of effort if you do not understand it If you understand it it is obvious seem to indicate some unfamiliarity with the idea of a mathematical proof The idea is to justify statements to a level where they are convincing to the intended reader Either you get it or you don t is not the way things are proved in mathematics Can you say what you disliked about the rewritten version I tried to keep the informal tone while not having all the terminological errors and vague claims of the modern proof Carl talk