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Whenever I Google to try to find an actual formal statement of the first incompleteness theorem (as opposed to all the oversimplified explanations that talk about "true but unprovable theorems" rather than theorems independent of the axioms), the definitions that don't just mention something like a system with "strong enough to do arithmetic on the natural numbers" mention a system with an "a sufficiently expressive procedure" for enumerating theorems, which is reminiscent of terminology used in explanations I've seen of Turing machines, and so I thought perhaps it meant that the incompleteness theorems apply specifically to Turing complete systems. However, when I posted about this on Quora I got responses saying that Turing completeness has nothing to do with it, whereas, on the askcomputerscience subreddit, I got responses saying that yes, the type of systems the incompleteness theorems apply to are mathematically equivalent to Turing complete models of computation.

So which is correct? Is a system with a "a sufficiently expressive procedure for enumerating theorems" just a Turing complete complete formal language/model of computation? If not, what exactly does that terminology mean?

To be clear, I understand that the first incompleteness theorem states only that a formal language with "a sufficiently expressive procedure for enumerating theorems" must be either inconsistent or incomplete (i.e. must either imply contradictions or contain theorems that are probably true in some models of the language and provably false in such models), so I don't need an explanation of what the theorem says, so much as clarification on what types of formal languages it actually applies to.

I did see this answer to a related question, which seems to imply that the types of languages/systems the incompleteness theorem applies to are systems that are "recursively axiomatized" systems, but it's not entirely clear what that means. Based on other simplified explanations I've seen of the the theorem, I guessing it refers to something like "a system sufficiently powerful to talk about itself" which would intuitively seems to match up with the concept of using Godel numbering to construct a statement like, "This theorem is unprovable", but that still seems rather vague and nonrigorous. Surely, there has to be a specific and rigorous definition/description of the sorts of systems the theorem applies to though, otherwise how would it be useful?

Mikayla Eckel Cifrese
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    There's nothing mysterious about the phrase "recursively axiomatized," although it is technical: a theory $T$ is recursively axiomatized if there is a set of sentences $A$ such that $(i)$ the things $A$ prove are exactly the things in $T$ (or are exactly the things $T$ proves, if we don't require theories to be deductively closed) and $(ii)$ the set $\hat{A}$ of Godel numbers of elements of $A$ is recursive, that is, there is a computer program which outputs $1$ when input an element of $\hat{A}$ and outputs $0$ when input a non-element of $A$. – Noah Schweber Aug 03 '23 at 22:27
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    A better term might be "computable," and there has been a terminology shift from recursive to computable starting in the early 90s. In particular, recursiveness in this sense has nothing a priori to do with self-reference, despite the name. The point is that there are theories of interest - like ZFC and PA, for instance - which are not finitely axiomatizable, but they're still "okay" since they are recursively/computably axiomatizable. – Noah Schweber Aug 03 '23 at 22:27
  • @NoahSchweber, OK, so if it's really about computability, was my instinct about the theorem applying only to Turing complete languages correct? – Mikayla Eckel Cifrese Aug 03 '23 at 22:46
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    @NoahSchweber: it is a prejudice from CS that recursion has something to do with self-reference. Recursion stems from a Latin word meaning "run back" and so a recursive computation (for a Roman) would just be one that runs back form $n$ to $n-1$ to $n - 2$ to ... $1$ (or $(0)$). Sadly, the change in terminology from "recursive" to "computable" seems to be pandemic. (The last thing that a Roman would have thought was that "recursio" meant running back to his house with something different in his hands. I stand ready for correction from any MSE classicists whose Latin is better than mine.) – Rob Arthan Aug 03 '23 at 22:54
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    @RobArthan, why is a change in terminology sad? Words change meaning all the time because people just start using them in different ways. As long as clear communication is maintained, I see no problem. – Mikayla Eckel Cifrese Aug 04 '23 at 02:14
  • @MikaylaEckelCifrese: clear communication is not maintained by the terminological change I am talking about. Mathematical terms generally do not change meaning all the time. If you start talking about groups, rings and fields as hens, swans and geese, you will not be understood. – Rob Arthan Aug 04 '23 at 07:47

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