TITLE: Where did this presentation of Godel's theorem appear?
QUESTION [11 upvotes]: This question was asked and bountied at MSE, with no response.

Many years ago I ran into the following proof of Godel's first incompleteness theorem
(here $T$ is an "appropriate" theory of arithmetic.):

First, we observe that Tarski's undefinability theorem can be slightly tweaked to say that for all $M\models T$, the theory of $M$ can't be the standard part of an $M$-(parameter-freely-)definable set. Next, by the usual representability arguments we have that every computable set is invariantly definable: if $X\subseteq\mathbb{N}$ is computable and $M\models T$ then $X$ is the standard part of a definable set in $M$. Now if $S$ were a computable satisfiable completion of $T$ we would get a contradiction by looking at $M\models S$. If we want, we can then replace "satisfiable" with "consistent" via the completeness theorem.

My question is:
Where did this argument appear?
I'm not asking whether this is a "genuinely different" argument (although that said, see here). Rather, I'm interested in its presentation in terms of invariant definability. This was a notion first introduced by Kreisel following Godel/Herbrand and subsequently studied by others (see e.g. 1, 2, 3). My vivid recollection is that the argument above appeared as a footnote in the first paper mentioned above, but it turns out it's not there after all.

REPLY [3 votes]: This argument is essentially similar to the argument of Mel Fitting in his article,  "Russell's paradox, Gödel's theorem" Chapter in book: Raymond Smullyan on self reference, 47–66, Outstanding Contributions to Logic, 14, Springer, Cham, 2017.
We had used this article as one of the readings in my Graduate Phil Logic seminar last term.
Fitting argues via Russell's paradox using Tarski's theorem on the non-definability of truth, in a manner that I find very similar to the argument you describe (although he doesn't use your invariant terminology, and perhaps you may regard his argument as not the same as what you intend). Namely, he argues that although provability is representable, truth is not, because of Russell's theorem, and so the incompleteness theorem follows.
Part of his point in that article is that this way of arguing avoids the need for explicit self-reference.