mongre at gmx.de
Wed May 18 03:01:07 EDT 2011
I only watched this once, and I didn’t bother with the questions. but I
do think FOMers have been too quick to jump at his (surely
deliberately!) outrageous misstatement of Gödel’s results and too
unwilling to think what he might mean.
There is an old tradition according to which mathematics has to be
*absolutely indubitable*. *We* can make mistakes in thinking we have a
proof when we haven’t really got one, but *mathematics* can’t make
mistakes: if we do have a proof then what we have proved is true (and
known by us a priori). (I go no further into the problems this raises.)
Genuine mathematical results have to be in some sense absolutely
Now think of the Hilbert programme in this context. Disputably
simplifying: Pi_1 statements are capable of truth or falsity
(‘truth-apt’ in the ugly philosophical jargon). They are true/false iff
provable to be such with finitary reasoning (which we can, simplifying,
take to be PRA, supposed to be indubitable). A proof in PRA that some
infinitary system is consistent would (by a familiar argument) show its
Pi_1 theorems to be (indubitably) true. So if we have it, for PA say, we
will thereby guarantee the indubitable truth of all the pi_1 theorems of
PA. Gödel shows us that we *can’t* have it.
Now what I take Voevodsky to be saying: we can coherently doubt the
consistency of PA. This seems obviously true, since if it weren’t, the
Hilbert programme for PA would have been incoherent. Further, although
there are proofs of the consistency of PA, for each such proof we can
doubt its cogency. (The examples Voevodsky considers are firstly the
obvious proof that the axioms of PA are true and the proof rules
preserve truth (formalized, this is the proof of Con(PA) in second-order
arithmetic) where it is possible to doubt whether we really understand
induction over arithmetically definable but non-recursive properties, or
Gentzen-style proofs where it is possible to doubt whether the ordinals
less than epsilon_0 (modulo some suitable notation) are really
well-ordered. This seems to me correct. (I don’t mean I *do* doubt these
things - I don’t - but one *can* coherently doubt them.) The consistency
of PA is not absolutely indubitable.
But of course if PA is inconsistent then anything is provable from it,
so it follows that proving something from PA doesn’t make it absolutely
Now that doesn’t worry *me* at all, because I don’t think that
mathematical theorems have to be indubitable. I’m confident (without
certainly) that PA (or for that matter ZF) is consistent (indeed true!),
and I’m happy to let my mathematical knowledge be fallible. But suppose
you’re more of a traditionalist, and you’re *not* happy with that.
I don’t think the point is that PA *might turn out* to be inconsistent.
Of course it might, and if it did, we would look at the induction axioms
used, try to establish what was peculiar about them, and then weaken PA
accordingly. That’s fantasy land, and is not going to happen.
Rather the point is that we want to know now with *absolute certainty*
that the results we’ve deduced from PA have to be true *even if PA is
inconsistent*. And then the claim was (and here I lost him) that by
considering rules of deduction other than the usual ones we could show
in some cases (not all) that the theorems would have to be true even if
the axioms were inconsistent. We would get a sort of Hilbert programme
I’d be amazed if such a programme were to go through, and we need to be
told exactly which sentences are truth-apt, and what their truth
consists in, but is there any reason why it’s obviously silly?
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