[FOM] The certainty of mathematical proof
Timothy Y. Chow
tchow at alum.mit.edu
Wed May 25 11:01:17 EDT 2011
Staffan Angere wrote:
>I tend to agree with Voevodsky that the question of whether PA is
>inconsistent is still open, even if I would not say that I doubt its
>consistency. One of the things Goedel showed was that it always will be
>open, unless we find an actual inconsistency in it - at least of the
>standards of proof required are such as to include taking the falsity of
>PA as a live possibility.
Let me present a fictitious dialogue, to try to convey to you what your
suggestion sounds like to f.o.m. experts. Instead of PA, which is overly
familiar, let me take the axioms for a DLOWFOLE (a dense linear ordering
without first or last element). A DLOWFOLE is a set together with a
binary relation "<" satisfying the following axioms:
1. For any x and y, exactly one of the following holds: x<y, x=y, y<x.
2. If x<y and y<z then x<z.
3. For any x<z there exists y such that x<y<z.
4. For any y there exist x and z such that x<y<z.
Are these axioms consistent? Let's eavesdrop on a conversation between a
mathematician and a skeptic.
Mathematician: Yes, the axioms are consistent, because Q, the set of
rationals with the usual order relation, is a DLOWFOLE.
Skeptic: That "proof" is unconvincing because it's so trivial. You're
taking for granted the existence of Q, but if you're going to do that, you
might as well just take for granted the consistency of the DLOWFOLE
axioms. Anyone with doubts about the consistency of the DLOWFOLE axioms
will likely have doubts about Q too, so your "proof" has no probative
Mathematician: The proof is trivial because the result is trivial. What
are you looking for, a complicated proof of a simple fact?
Skeptic: I'm looking for a proof from first principles that doesn't just
take the existence of Q for granted.
Mathematician: What do you have against Q? It's a basic object of
mathematical study. If you don't accept Q then you're tossing out huge
swaths of mathematics.
Skeptic: You're not answering my question. I'm beginning to think that
one *can't* prove the consistency of the DLOWFOLE axioms. Don't Goedel's
theorems tell us that we can't prove consistency statements, but only
inconsistency statements? If so, then perhaps the consistency of the
DLOWFOLE axioms will remain forever an open problem.
Mathematician: An open problem?! You must be joking. It's a total
triviality that Q is a DLOWFOLE. I'd hesitate to even call it a
"problem," let alone an *open* problem. Now, suppose you were to ask
whether there are any countable DLOWFOLEs that aren't isomorphic to Q.
That's an interesting problem, though it turns out that it was settled a
long time ago---there aren't any. Do you want to hear the proof?
Skeptic: No, no...I'm sure it's a nice proof but that's not what I'm
interested in. I still think it's conceivable that one day someone
might demonstrate an inconsistency in the DLOWFOLE axioms. Isn't it
Mathematician: Conceivable? You mean like it's conceivable that you're a
brain in a vat, or that it's conceivable that the world began yesterday
with all our memories in place?
Skeptic: No, not that extreme. I mean that it seems to me that you're
making a philosophical assumption that Q exists in your "proof" of
consistency. Isn't that right? Isn't that a philosophical assumption
rather than a mathematical assumption? If so then maybe it's wrong and Q
doesn't exist and the DLOWFOLE axioms are inconsistent.
Mathematician: Uh...I don't know how to answer that...
Skeptic: Oh well, never mind. Anyway, what you've said makes me lean
towards the view that the consistency of the DLOWFOLE axioms is an open
problem---not that I doubt that they're consistent, of course. Thanks for
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