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.

Nelson has withdrawn his claim. Link.

On the FOM list, he writes:

More here, including a comment by Terence Tao.

Specifically, Tao's comment:

He's such a glorious mathematician. <3

He gave a more detailed comment on the n-Category Café.

FTL neutrinos and now a proof of inconsistency in Peano Arithmetic? What next?

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... (read more)

And here I thought there wasn't anything besides

cI'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.

Would you bet at 99-to-1 odds that I will not win the lottery tomorrow?

Assume for a second that FTL communication is possible and that PA is inconsistent. How could this possibly influence a proof of AI

friendlinessthat has been invented before those discoveries were made and how can one make an AIprovablyfriendly given other possible fundamental errors in our understanding of physics and mathematics?This aspect is very interesting:

... (read more)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.