[FOM] Strict reverse mathematics of incompleteness.

GARETH JOHN YOUNG g.young.3 at research.gla.ac.uk
Sun Jun 19 19:53:36 EDT 2011

That should be 'Voedovsky'. Apologies.

Sent: 20 June 2011 00:46
To: Foundations of Mathematics
Subject: RE: [FOM] Strict reverse mathematics of incompleteness.

Dear All,

I've been thinking similar things to Richard Heck to the effect that it's not at all obvious how any of the previous discussion should lead us to believe that Con(PA) is false. Lack of a proof of P (indeed, proof that there is no proof of P) is no reason whatever to think that P is false. The most charitable reading I can put on Voedevsky's talk is that Goedel's theorems show that there is no *epistemically interesting* proof of the consistency of PA. That is, any proof of PA will be in a system whose consistency, if we are in a context where we doubt Con(PA), should equally be in question. It's not obvious to me that Goedel's theorems are especially exciting here since, even if PA could prove its own consistency, that doesn't show it's consistent (it could equally do that if it was inconsistent). Goedel's theorems don't seem to me to obviously make matters different, epistemically speaking.
The moral we should draw from these considerations is not that we should think systems like PA or Q are inconsistent, or even that we don't *know* these systems are consistent, but at the very most that we shouldn't regard their consistency as an absolute certainty.

Daniel Mehkeri's argument against the skeptic seems to go thus: We need at least Q to formulate the incompleteness theorems. But if we doubt even Con(Q) (which we ought to, if we doubt Con(PA)), then why give the  argument from the incompleteness theorems any credence if we need at least Q to prove them? I take it the claim is that the skeptic's claim is undermined here because there is no proof of the incompleteness theorems which the skeptic is entitled to claim as independantly kosher.
Alas, I don't think this will stick. The skeptic can simply claim that, for pretty much any interesting system you like, suppose it's consistent, then the incompleteness theorems hold. If it's inconsistent and we can trivially prove the incompleteness theorems, it hardly seems like a victory for the champion of Con(Q).

Like I said, we can't prove Con(PA) in any system of whose consistency we have independent certainty. So what? We have excellent reasons to believe Con(PA) (not least the lack of planes falling out of the sky, cited by Voedevsky in his talk) and none that I'm aware of to believe its negation. So we're justified in believing Con(PA) (absent some argument from Voedevsky), but not with Cartesian certainty. Our believe in Con(PA) doesn't seem to me to be much worse off than loads of other beliefs, like the one we share in the existence of the external world which is without any kind of ultimate proof but which we nonetheless take ourselves to know.

I also don't know why Voedevsky suggested that, if Con(PA) is false, finitistic proofs would be the way to go. Does he have any reason to think that finitistic systems should be any more likely to be consistent than standard PA?

The discussion has puzzled me, so if I've missed something obvious, it would be helpfu to mel if someone could point it out.

Best wishes,

Gareth Young
From: fom-bounces at cs.nyu.edu [fom-bounces at cs.nyu.edu] On Behalf Of Daniel Mehkeri [dmehkeri at gmail.com]
Sent: 18 June 2011 19:15
To: fom at cs.nyu.edu
Subject: [FOM] Strict reverse mathematics of incompleteness.

Are there any reverse mathematical results relevant to Gödel's
incompleteness theorems? It has been mentioned that even Robinson's Q
does not prove its own consistency. But I am also wondering about
something orthogonal to that: not just what is required of a formal
system for the incompleteness theorems to apply to it, but also what is
required of a formal system to prove the incompleteness theorems.

This would be at the level of "strict" reverse mathematics as Friedman
calls it. I am vaguely aware that encoding of syntax becomes an issue
here, and so the question may not even be precise enough as stated.
Unfortunately I don't know how to state it more precisely than that.

This is relevant to the Con(PA) discussion of course. Briefly, if the
incompleteness theorems cause us to doubt Con(PA), then why not doubt
Con(Q)? But if Con(Q) is in doubt, then why believe the incompleteness
theorems? So it might be useful to have a better idea of what the
incompleteness theorems require.

Daniel Mehkeri

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