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Löb's theorem states that a mathematical formula which (metamathematically) asserts its own provability is indeed provable. More precisely, if a sufficiently strong system T (e.g. PA or ZFC) proves $Prov_T(\varphi) \rightarrow \varphi$, then T actually proves $\varphi$.

If one analyzes the proof, one sees that the proof works for any formalized provability predicate satisfying certain conditions (that are listed in the Wikipedia page). My question is the following:

Has anyone tried to prove some natural mathematical statement $\varphi$ using Löb's theorem, by constructing a provability predicate for which one has $Prov_T(\varphi) \rightarrow \varphi$ and which satisfies the necessary conditions?

I am aware that it sounds like an impossible mission. But surely someone must have tried to "reverse engineer" a predicate at least for certain number-theoretic statements?

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  • $\begingroup$ I think it's worse than impossible - how would you propose to even begin? $\endgroup$ – Noah Schweber Jan 15 '17 at 0:56
  • $\begingroup$ @NoahSchweber: Well, we have the Gödel number of $\varphi$ and want to come up with a property which, if holds at that number, implies $\varphi$. This part I think may be handled. Box distributivity doesn't seem too imposible as well. We want to tweak the previous property so that when it holds for some Gödel number (of "A implies B"), it holds for the Gödel number (of B) defined by a cofinal segment of exponents in the prime factorization whenever it holds for the Gödel number (of A) defined by a certain initial segment of exponents in the prime factorization. $\endgroup$ – Burak Jan 15 '17 at 1:13
  • $\begingroup$ @NoahSchweber: For the other essential properties of the box operator we need, I was hoping that there may be some really really clever guy! $\endgroup$ – Burak Jan 15 '17 at 1:14
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I wouldn't expect that to yield anything useful.

Löb's theorem is fundamentally a negative result. What it does is to pull the rug under this otherwise intriguing proof strategy:

I want to prove $\psi$ (within $T$), but that seems to be hard. Instead I will use some kind of non-constructive meta-reasoning (that is formalizable in $T$) to show that a proof of $\psi$ must exist (even though I may not learn how that proof works), and then also prove $\operatorname{Prov}_T(\varphi)\to \varphi$ for a class of statements that includes my $\psi$. Then, combining those two parts, I will have proved $\psi$.

Löb's theorem tells us that this strategy is fundamentally flawed, because if you manage to complete the "easy" part $\operatorname{Prov}_T(\varphi)\to \varphi$, then you actually don't even need the non-constructive meta-reasoning that looked like the clever shortcut at first.

The lesson we learn is that proving $\operatorname{Prov}_T(\varphi)\to\varphi$ is harder than it looks -- you can't prove this for some nice general syntactically recognizable class of $\varphi$s unless it's because every $\varphi$ in that class is true (in which case having a particular proof in your hands is not very exciting).

In other words, proving $\operatorname{Prov}_T(\varphi)\to\varphi$ inescapably needs to involve reasoning about the particular thing that makes $\varphi$ true, rather than just things that would make us trust a separately given proof of $\varphi$. And if you're doing that kind of reasoning anyway, the argument goes, you might as well be proving $\varphi$ directly. That doesn't look like the makings of a shortcut.

(There are some "unnatural" exceptions to this where $\varphi$ is something you construct using the fixpoint theorem with the express goal of making $\operatorname{Prov}_T(\varphi)\to\varphi$ necessarily true. But that's explicitly not what you're asking).

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