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Take any formal system $S$ that has a proof verifier program and interprets TC or PA$^-$. $ \def\imp{\Rightarrow} \def\con{\text{Con}} $

Then the incompleteness theorems show that $S$ does not prove $\con_1(S)$, where the subscript denotes that it is based on a particular encoding of a particular proof verifier of $S$ into $S$. $S$ also does not prove $\con_2(S)$, for another choice of proof verifier and encoding. But...

Does $S$ always prove $( \con_1(S) \imp \con_2(S) )$ (regardless of the two choices)?

I think the answer is no, because it seems I need $S$ to support induction to establish such a proof, namely I seem to need $S$ to interpret either PA or TC+I (where I is an induction schema). So...

If my guess is wrong, how do we show that $S$ proves the equivalence?

If my guess is correct, what are two choices of $\con_1$ and $\con_2$ that witnesses it?

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This is tricky. It is equivalent to the question of whether standard Gödel sentences for $S$ produced by two different encodings will be provably equivalent in $S$.

Suppose we define a Gödel sentence to be a fixed point of a "not-provable-in-$S$" predicate, then choose a coding which determines the details of the not-provable predicate. Then the fixed points of this one not-provable predicate will be interderivable. That's elementary, well-known, and in lots of textbooks.

But that doesn't help with your question, which comes to: if we vary the not-provable predicate by varying the coding, will the fixed ponts as we vary the predicate still stay interderivable? Presumbably there will have to be some restrictions on what counts as an acceptable coding: but I don't know what the restrictions are. Nor do the usual (or not-so-usual) textbooks tell us, again as far as I know. Which is why I asked a similar question at mathoverflow: the response was "I don’t think there are any useful criteria known that would guarantee the provable equivalence of two proof predicates that would not beg the question."

I too would be delighted to know if we can do better than that!

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  • $\begingroup$ Can you at least confirm my claim that if $S$ supports induction then we in fact have provable equivalence? My reasoning (but I was a bit lazy to formalize it) is that we ought to be able to prove within S by induction that if Con1(S) is false with proof length n then Con2(S) is false. And thanks for the link; I totally did not know that you had asked this before on MathOverflow! =) $\endgroup$ – user21820 Dec 10 '17 at 16:35
  • $\begingroup$ I doubt whether the claim is true in general for PA (which has induction), unless you put some constraints on allowable codings. What constraints do you have in mind? $\endgroup$ – Peter Smith Dec 10 '17 at 16:38
  • $\begingroup$ Hmm.. I had in mind that we start with any program that implements a proof verifier $V$ such that $V(p,x)$ always halts and outputs "yes" if $p$ is a proof of $x$ and "no" otherwise. This would vary drastically depending on the proof-style, and that also means that my attempt is doomed. We need some uniform computable interpretation of TC+I or PA with computable inverses... where "uniformly interprets" is like as defined here. Oh that seems insufficient too unless $S$ is simply TC+I or PA itself, which seems like "trivial" then. $\endgroup$ – user21820 Dec 10 '17 at 16:45
  • $\begingroup$ By the way, I wrote a recent post sketching a computable procedure for extending any given $Σ_1$-sound system to one that is still $Σ_1$-sound but $Σ_3$-unsound. As I ask in my comment, can this be improved? I gave a non-constructive argument in my answer, but I doubt it can be made constructive. Any ideas? $\endgroup$ – user21820 Dec 13 '17 at 13:36
  • $\begingroup$ Hmm do you think I should post my other question on MO? $\endgroup$ – user21820 Dec 16 '17 at 9:20

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