Incompleteness Theorems for Limit-Computable Formal Systems

Question

Godel’s First and Second Inconpleteness Theorems are about Peano Arithmetic, but their punchlines are respectively that “Any sufficiently expressive computable formal system cannot be complete and consistent” and that “Given a computable emulation of the proof theory of such a system inside itself, no proof consistency can be given, if the system is consistent”. In mathematics we like certainty (or something like it, for those of you with less objective dispositions), so this is a problem. However, practical considerations do not always need certainty. In this spirit, we might not require a computable formal system, but merely a limit-computable one. By this I mean that there is a theoretical machine $m$ which takes pairs of sentences $s$ of the system and natural numbers $n$ as inputs and outputs $0$ or $1$ assuming it halts. Provability of $s$ in the system is that $$\lim_{n\to\infty}m(s,n)=1$$ which just says that, for all but finitely many $n$, $m$ outputs $1$ on input $(s,n)$. In this way, if we accept the system, we can at least be sure that it will eventually tell us the right answer. Are there analogues of the Incompleteness Theorems for such systems? Or similar results?

Answer 1

I'm going to focus on the first incompleteness theorem below, except for the final paragraph.

EDIT: the material here can all be found in Salehi/Seraji, Godel-Rosser's incompleteness theorems for non-recursively enumerable theories (although I was unaware of that paper when I wrote this answer). Additionally, note that Ishii, A note on the first incompleteness theorem claims a result contradicting the following, but that paper is incorrect and the error is addressed by Salehi/Seraji.

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There are two problems with the theory you describe. The first is that, as you observe, it is not actually computable - and as finite entities, we can't really make that much use of limit computability. The second issue is that it is unsound: it is complete and consistent, but proves false sentences.

When we take soundness into account we see that there is indeed a version of the incompleteness theorem which "relativizes" to limit computation and beyond. Specifically, for each $n>0$ we have:

($\mathsf{GIT_n}$): Suppose $T$ is a consistent theory of arithmetic which contains Robinson arithmetic, is ${\bf 0^{(n-1)}}$-computably axiomatizable, and is $\Pi^0_n$-sound. Then $T$ is incomplete.

(As usual this can be generalized substantially, but let's focus on this version for simplicity.)

A couple quick comments:

• "${\bf 0^{(n-1)}}$" is the $n-1$th iterate of the Turing jump. The case $n=1$ corresponds to computably axiomatizable theories (= the original GIT).

• Using Craig's trick, we can replace "${\bf 0^{(n-1)}}$-axiomatizable" by "$\Sigma_n$-axiomatizable" (the usual version of Craig's trick is that computably axiomatizable and c.e. axiomatizable are equivalent, but it generalizes).

• "$\Pi^0_n$" is a particular class of sentences in the language of arithmetic - see here. "$\Pi^0_n$-soundness" is the property of only proving true $\Pi^0_n$ sentences. Note that when $n=1$ this is a redundant hypothesis, since every theory containing Robinson arithmetic proves all true $\Sigma^0_1$ sentences and so cannot prove any false $\Pi^0_1$ sentences without being inconsistent; this is why this additional hypothesis doesn't appear in the usual statement of the first incompleteness theorem.

• The two notions above are related to each other and to (a generalization of) limit computability: see Post's theorem and Shoenfield's limit lemma. In particular, we have "computable = $\Delta^0_1$ = ${\bf 0^{(0)}}$-computable" and "limit computable = $\Delta^0_2$ = "${\bf 0^{(1)}}$-computable."

The $\mathsf{GIT_n}$s above can be proved by relativizing any of the computability-theoretic proofs of the usual incompleteness theorem; a useful trick here is that if $T$ satisfies the hypotheses of $\mathsf{GIT_n}$ then so does $T$ + the set of true $\Sigma^0_n$ sentences (this is a good exercise). Proof-theoretic arguments can also be used, but their relativizations are in my opinion less immediate.

[The second incompleteness theorem poses an interesting new difficulty, on the other hand; see an earlier edit of this question in conjunction with Andreas Blass' comment below.]