Saturday, 1 October 2011

Nelson withdraws his claim

Edward Nelson has retracted his claim of having a proof of the inconsistency of PA. Replying to Terence Tao, he says:
One wonders if he will try to pursue the idea of proving the inconsistency of PA, or if the mistake spotted by Tao is too crucial to be repaired. By the way, here is a cool post on the whole issue over at Godel's lost letter and P = NP (via Ole Hjortland).

But anyway, until further notice, PA is still consistent; we can now go enjoy the weekend in peace.

(And thanks to John Baez for posting on Nelson's withdrawal at G+.)

UPDATE: Here is the message that Nelson just sent around at FOM:
Terrence Tao, at and independently Daniel Tausk (private communication) have found an irreparable error in my outline. In the Kritchman-Raz proof, there is a low complexity proof of K(\bar\xi)>\ell if we assume \mu=1, but the Chaitin machine may find a shorter proof of high complexity, with no control over how high.

My thanks to Tao and Tausk for spotting this. I withdraw my claim.

The consistency of P remains an open problem.

Ed Nelson
This seems to be a good example of what J. Azzouni has described as the 'uniqueness' of mathematics as a social practice: in just a few days, a consensus has emerged as to what was wrong with Nelson's purported proof, including Nelson himself. I cannot think of any other field of inquiry where consensus on a substantive, serious issue/challenge would emerge with the same swiftness. There really is something special about mathematics...


  1. My congratulations to Daniel Tausk (and Rodrigo Freire) at IME/USP for so carefully and quickly scrutinizing Nelson's pretensions.

  2. Another interesting observation is that this 'swift debate' will have raised Nelson's standing in the mathematics community even though he was wrong. When cranks turn up and try to keep pointless debate going; the debate without progress is what is used to identify them as cranks.

  3. I agree with Roger. It's an important point.

  4. Yes, there are many interesting semi-sociological aspects in these events. I'm hoping to find the time to write a post contrasting them with the immediate reception of Godel's incompleteness results.

  5. Good idea!
    A few months ago, I was going to write a post on exactly this topic. I have a paper from 1999, whose first theorem, Theorem 1, is incorrect. (Using a side condition, it can be fixed though - phew.) I'm quite happy to tell people it's not quite right (have had a corrigendum on my webpage in Edinburgh for years). Plus, in section of 4 of the same paper, I have a theorem which is right, but whose generalization to another case I missed, and which vitiates the argument I give in that section!. Notre Dame logician Tim Bays has a good analysis of this in a 2009 Mind paper, and my reply was basically: "You're right, Tim. I've made a mistake".

    Personally, I can't quite fathom why someone working in a technical or semi-technical area would refuse to confess to an error, when to any reasonable expert it is clear. It's just simple intellectual honesty. If you goof up, own up! It's no big deal (everyone goofs up!), and that's why Roger's comment is bang on.

    (Once in logic teaching, I didn't check carefully enough, and formulated an annoying substitution axiom incorrectly in my lecture notes. Only a couple of months later did I notice the mistake.)

    So, the real measure is how one responds when others point out a goof-up, whether a minor goof-up in your lecture notes, or whether the claim is a mega-claim and the problem is very hard to discern (as in the P =/= NP paper last year).

    Unfortunately, I do know of some extremely wretched examples of refusal to own up - but from philosophers.