[FOM] Consistency of Robinson arithmetic

[Richard Heck]

On 05/18/2011 04:48 PM, Timothy Y. Chow wrote:
> The current discussion about the consistency of PA makes me wonder: Does 
> Voevodsky or Nelson doubt the consistency of Q?  Some of the vague 
> objections that I sketched in my previous post would apply just as well to 
> Con(Q) as to Con(PA).  Getting a straight answer to this question could 
> sharpen the discussion, by clarifying whether PA is just being used as an 
> arbitrary placeholder or whether there is something specific about PA that 
> is (allegedly) problematic.
>
I was going to raise this issue later. (I'm correcting proofs at the
moment and so have little time for anything that counts as fun.)
Certainly the 'circularity' worry you mentioned in an earlier post has
to apply as much here as anywhere.

That said, it is worth distinguishing different sorts of consistency
proofs. Consider, for example, the usual Tarskian proof of the
consistency of T, for arbitrary T. We can prove a quite general theorem
to the effect that, if you start with some theory T and then add a
syntax (in a separate sort) and a semantic theory for the language of T,
with \Sigma_1 induction in the syntax, and with semantic notions
permitted in the induction axioms (sorry that was so long), then the
resulting theory proves Con(T). That kind of argument, though it does
count as a proof, has not probative force and is correctly described, as
it often is, as "trivial".

But not all consistency proofs are like that. Certainly Gentzen's isn't,
and I'd personally not count the proof of Con(PA) in ZFC as trivial in
that way (though I might agree that it isn't very informative).

For what it's worth, I think it's a very interesting question whether we
know that PA is consistent. Not so much because I don't think we do, as
because recent work in epistemology may well throw light on the question
and why different people have different views. Your reference to
context-dependence in your previous message was most notable in that
respect, as contextualist views about knowledge are still quite common,
and even so-called "subject-sensitive invariantism" might allow us to
make similar remarks.

Perhaps the most intriguing possibility, to me, however, lies in work on
what people call "transmission failure". These are cases where one knows
that P, has a good (logical) argument from P to Q, and yet it seems as
if one can't (come to?) know Q on that basis. A classic sort of
(putative) example is: I know that that thing in the cage over there is
a zebra; if it's a zebra, then it isn't a cleverly disguised mule; so I
know that it isn't a cleverly disguised mule. Here, it seems as if one's
belief (even knowledge) can't support the claim that the thing isn't a
cleverly disguised mule. It's as if there's a sort of epistemic circularity.

Whether that is a good example, or whether there are any such examples,
is of course controversial. (This is philosophy!) But someone who was
worried about some kind of circularity in consistency proofs might
appeal to this sort of idea. It's especially illuminating, it seems to
me, as regards the "trivial" proofs we learned from Tarksi.

Back to my proofs....

Richard Heck

-- 
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Richard G Heck Jr
Romeo Elton Professor of Natural Theology
Brown University