In case you missed a recent post by Richard Heck on the FOM list, let me recommend two pieces linked there.
First, at the Saul Kripke’s memorial conference in May, Dan Isaacson gave a talk on “Kripke on Gödel Incompleteness” where he discussed six of Kripke’s papers around and about incompleteness. You can download Isaacson’s interesting paper here. His §5 is very brisk on Kripke’s very short 2020/21 paper “Gödel’s Theorem and direct self-reference”: I also say something about Kripke’s note in the two-page Appendix to the second edition of GWT.
The construction Kripke describes in “Gödel’s theorem and direct self-reference” was independently discovered and published by both Albert Visser [see his handbook paper on the liar] and myself [in “Self-Reference and the Languages of Arithmetic”]. There’s a footnote in Kripke’s paper about this, but, frankly, I don’t see what’s new in that paper of Kripke’s (though it may well be that he had all this before Visser and I did, I don’t know). Once you have the construction, you can obviously use it for any purpose for which Gödel numberings have been used, e.g., the proof of the incompleteness theorem.
There has been quite a lot of recent work, in fact, on the details of Gödel numberings and intensionality that can arise because of differences between them. For example, the mentioned paper of mine describes a theory of truth that is consistent given ‘typical’ Gödel numberings but inconsistent if you use a Gödel numbering that permits the sort of ‘direct’ self-reference at issue in Kripke’s discussion. Other papers in this tradition include Halbach and Visser’s two papers on “Self-Reference in Arithmetic”, Grabmayr and Visser’s “Self-Reference Upfront”, Grabmayr’s “On the Invariance of Gödel’s Second Theorem With Regard to Numberings”, and Grabmayr, Halbach, and Ye’s “Varieties of Self-Reference in Metamathematics”.
I confess most of those titles are new to me, as I’ve not been keeping up with this literature recently, and given the authors the pointers are likely to be well worth following up.
Something I have read with enjoyment is Heck’s own overview of some of these ideas and related matters, written for a forthcoming handbook on the liar paradox. Written with characteristic clarity here is a version of the paper.