Yes, that’s fair. Goedel’s incompleteness theorem is about Peano Arithmetic (or Robinson arithmetic or its Sigma_1 fragment or true arithmetic or the corresponding theories of hereditarily finite sequences or what have you) the way Taylor’s theorem is about hyperbolic trigonometry (or elliptic integrals or the Gamma function or Gaussian distributions). Those are all specific areas yielding instances to which the theorem is applicable, but the theorem is not fundamentally about any of those in particular, and it would be odd to present it in terms of them.
Goedel’s theorem is not even fundamentally about just computable theories [as seen by its alter ego, Tarski’s indefinability theorem]. (For that matter, I daresay the near-universal presentation of GIT1 in terms of incompleteness is an accident of history, setting an odd focus. But I digress). For example, it tells us that an axiomatization of true arithmetic (all the true first-order statements in the language of PA) cannot even be given by a program equipped with an oracle for the halting problem + an oracle for the halting problem for machines equipped with an oracle for the halting problem + an oracle for … .[sup]1[/sup] This, of course, goes well beyond plain computability.
That having been said, computable theories are of course the particular case of most significant interest and clearest foundational importance, so I understand why people would want to present GIT only in terms of the restrictions it places on computably axiomatizable theories. And since interpreting the theory of computable functions within the finitary fragment of set theory or the theory of hereditarily finite sequences or such things is rote and straightforward, even if tedious, I couldn’t mind too much if people continually presented GIT in terms of its implications for extensions of natural systems of finite set theory. [And indeed, at the time of the 1931 paper introducing GIT, still five years before Turing machines or Post machines or the lambda calculus (a lot happened in 1936), those mechanics of defining and dealing with computable functions which are nowadays rote and straightforward were anything but[sup]2[/sup], and a major part of the paper was devoted to explaining these in detail, a fine accomplishment of Goedel’s at the time (given that no one had even set out a definition of the computable functions before). So, again, I can understand why one would continue to connect GIT to such matters of computability, given the seminal historical connection.].
But the final link that now somehow gets emphasized so much, that such systems of finite set theory can furthermore be interpreted within PA (and even Robinson Arithmetic) with devious enough coding, so that whenever one hears GIT1 mentioned, it is always with the refrain “for systems as strong as Peano Arithmetic”, “for systems as strong as Peano Arithmetic”, “for systems as strong as Peano Arithmetic”!. Well, this last part is a parlor trick of vanishing significance, an utter hack, analogous to an amusing programming accomplishment which may deserve an “Ooh, that’s a clever trick”, but doesn’t belong in any kind of code meant to be read and maintained. Even Goedel barely cared about it and only mentions the possibility at the very last minute, tossed-off in a brief digression into number theory near the end of the 1931 paper; in fact, his focus throughout the paper is actually on a system far stronger than PA, and even beyond first-order logic itself (specifically, the typed higher-order logic of Principia Mathematica, though mentions are also given to ZF, NBG, and the Peano axioms for the successor function + direct mechanisms for defining functions via simple recursion). Presenting GIT as tied at a fundamental level to this particular convoluted hack (feeding the establishment of arbitrarily long arithmetic sequences of coprimes into the Chinese Remainder Theorem to observe that finite sequences of naturals can be specified as the residues modulo some constant of some arithmetic sequence) is a little ridiculous, isn’t it? Particularly when this is detour is so often invoked even to illustrate the applicability of GIT1 to theories which can manipulate finite sequences much more directly (e.g., ZF).
Anyway, I forgot what I was going to do in the “GIT is not about arithmetic!” rant I was originally planning, but I’ve probably covered most of it here anyway.
1: In fact, in its original formulation, this is all GIT1 tells us, that this one very particular theory is very complex; it wasn’t till five years later that Rosser generalized GIT1 to the now familiar result concerning arbitrary (possibly consistent but false) extensions of PA
2: As an example of how things which were still being discovered and inobvious at the time are now immediate and mundane, the result of Rosser’s which took five years to see (like I said, a lot happened in 1936) was essentially nothing beyond the realization that computer programs can be computably simulated in parallel (so that if two ), though he did not cast it in these terms and it was also not clear at the time that this was all it amounted to.