On the FOM list, he writes:
Terrence Tao, at http://golem.ph.utexas.edu/category/2011/09/ and independently Daniel Tausk (private communication) have found an irreparable error in my outline. (...)
(...) I withdraw my claim.
The consistency of P remains an open problem.
Specifically, Tao's comment:
I have read through the outline. Even though it is too sketchy to count as a full proof, I think I can reconstruct enough of the argument to figure out where the error in reasoning is going to be. Basically, in order for Chaitin's theorem (10) to hold, the Kolmogorov complexity of the consistent theory T has to be less than l. But when one arithmetises (10) at a given rank and level on page 5, the complexity of the associated theory will depend on the complexity of that rank and level; because there are going to be more than 2^l ranks and levels involved in the iterative argument, at some point the complexity must exceed l, at which point Chaitin's theorem cannot be arithmetised for this value of l.
(One can try to outrun this issue by arithmetising using the full strength of Q_0^*, rather than a restricted version of this language in which the rank and level are bounded; but then one would need the consistency of Q_0^* to be provable inside Q_0^*, which is not possible by the second incompleteness theorem.)
I suppose it is possible that this obstruction could be evaded by a suitably clever trick, but personally I think that the FTL neutrino confirmation will arrive first.
He's such a glorious mathematician. <3
I have devised a little proof of inconsistency of the Newtonian mechanics, years ago.
http://critticall.com/alog/Antinomy_inside_mechanics.pdf
Can you spot the error?
So by now Nelson's outline has been challenged by the formidable Terry Tao, and Nelson (himself formidable!) has responded to this challenge and isn't budging. Link.
The FTL thread has attracted many confident predictions about the ultimate outcome. But this one hasn't. Is this because people find the subject less interesting? Or because they are less confident?
For what it's worth, here's the timeline of my thoughts/beliefs, in silly internal-monologue form. Maybe the numbers shouldn't be taken too seriously, and I'm not trying to bait anyone into bett...
And here I thought there wasn't anything besides c I'd bet on at 99-to-1 odds.
Two! Two things in the universe I'd bet on at 99-to-1 odds. Though I'm not actually going to do it for more than say $2k if anyone wants it, since I don't bet more than I have, period.
Assume for a second that FTL communication is possible and that PA is inconsistent. How could this possibly influence a proof of AI friendliness that has been invented before those discoveries were made and how can one make an AI provably friendly given other possible fundamental errors in our understanding of physics and mathematics?
This aspect is very interesting:
...Qea. If this were normal science, the proof that P is inconsistent could be written up rather quickly. But since this work calls for a paradigm shift in mathematics, it is essential that all details be developed fully. At present, I have written just over 100 pages beginning this. The current version is posted as a work in progress at http://www.math.princeton.edu/~nelson/books.html, and the book will be updated from time to time. The proofs are automatically checked by a program I devised called qea (for quod est absurdum
Here's a summary and discussion of the affair, with historical comparison to the Gödel results and their reception (as well as comments from several luminaries, and David Chalmers) on a philosophy of mathematics blog whose authors seem to take the position that the reasons for consensus in the mathematical community are mysterious. (It is admitted that "arguably, it cannot be fully explained as a merely socially imposed kind of consensus, due to homogeneous ‘indoctrination’ by means of mathematical education.") This is a subject that needs to be discussed more on LW, in my opinion.
We've discussed Edward Nelson's beliefs and work before. Now, he claims to have a proof of a contradiction in Peano Arithmetic; which if correct is not that specific to PA but imports itself into much weaker systems. I'm skeptical of the proof but haven't had the time to look at it in detail. There seem to be two possible weakpoints in his approach. His approach is to construct a system Q_0^* which looks almost but not quite a fragment of PA and then show that PA both proves this system's consistency and proves its inconsistency.
First, he may be mis-applying the Hilbert-Ackermann theorem-when it applies is highly technical and can be subtle. I don't know enough to comment on that in detail. The second issue is that in trying to show that he can use finitary methods to show there's a contradiction in Q_0^* he may have proven something closer to Q_0^* being omega-inconsistent. Right now, I'm extremely skeptical of this result.
If anyone is going to find an actual contradiction in PA or ZFC it would probably be Nelson. There some clearly interesting material here such as using a formalization of the surprise examiation/unexpected hanging to get a new proof of of Godel's Second Incompleteness Theorem. The exact conditions which this version of Godel's theorem applies may be different from the conditions under which the standard theorem can be proven.