`  :You said, ``But even [CBM] doesn't pretend that there is any inaccuracy in the proof anymore.`` I wish that you would avoid speaking for me. I posted this comment exactly to outline some problems with the text you added other than its originality.   :The proof you added is not complete, as it does not prove the arithmetization results that you consider ``easy to prove``; the proof you added is about as complete as the proof sketch already in the article. The text you added is not inaccurate in the sense of proving a false result, but it does use terminology in an inaccurate way. For example, as I said in the comment linked above, it is not at all obvious that a proof method known to Kleene in the 1950s is a ``modern`` proof.  :Finally, your claim ``all the current textbooks use Turing machines to prove the incompleteness theorems.`` is false, even if it is interpreted broadly to mean that the proof goes via the T predicate. For example, the proof in Hinman's Fundamentals of mathematical logic (2005) uses the approach via the diagonal lemma. — Carl ( ·  `