Hofstadter's actual thesis/intent with the book, and what he has to say about Gödel numbering (note: the book goes over Gödel's theorem and some implications, but Hofstadter's real interest is less in Gödel's theorem than in the proof technique—Gödel numbering—that Gödel used, because he is tickled by the self-reference), or Escher or Bach for that matter, are not really what make the book most worthwhile. What matters is the marriage of form and content, the perfect puns that seem to be set up several hundred pages apart, the way you'd be reading another dialogue before suddenly realizing that the lines are spelling out an acrostic, the satisfaction of things working themselves out perfectly (even the puns) (a feeling similar to the satisfying tying-together of the plots of certain novels, some of Wodehouse's included), and so on.
Very few authors can write about technical topics in a way their personality shines through. (Knuth is another IMO, except with the personality staying in second place to the technical content, while with Hofstadter it's the other way around!)
(Or non-technical writing: did you know about Hofstadter's translation of Eugene Onegin? He learnt Russian just to be able to read and translate that poem (see https://archive.nytimes.com/www.nytimes.com/books/97/07/20/r... https://www.tralalit.de/en/2018/07/25/a-tale-of-two-or-so-tr... https://www.york.ac.uk/depts/maths/histstat/pml1/onegin/ for the story, and https://archive.nytimes.com/www.nytimes.com/books/99/09/12/r... for the very negative NYT review) and the result is amazing for exactly the reasons the reviewer complains about: yes it can't stand alone and it's best read alongside a "real" translation say Falen's, but throughout you can see Hofstadter having fun with his word games, and somehow his personality shines through even in just the way he's choosing to translate Pushkin; it's so amazing how he manages to have so many things just work out. “Till his creditors cast their nyet's”)