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 betting 200 grand against my 10 dollars, but I'd be interested to hear what people have been thinking as (or if) they've been following this.
A few days ago: "Contrary to almost everybody, I don't think there is any reason to believe PA is consistent. Someone, e.g. Nelson or the much younger Voevodsky, might show it is inconsistent in my lifetime." P approx 5%.
Nelson's announcement: "Holy shit he's done it." P approx 40%
Tao's challenge: "Oh, I guess not, poor guy how embarrassing." P back down to 5%
Nelson neither ignores nor caves to Tao: "Beats me, maybe he's done it after all." P now maybe at 10%.
I've put up on Prediction Book a relevant prediction. The main reason I've given only a 75% percent chance that it may take a lot of time to study the matter in detail.
If you are putting 10% on the chance that Nelson's proof will turn out to be correct, then 10% seems way too high for essentially reasons that benelliot touched on below. My own estimate for this is being correct is around 0.5%. (That's sort of a meta estimate which tries to take into account my own patterns of over and underconfidence for different types of claims.) I'd be willing to bet 100$ to 1$ which would be well within both of our stated estimates.
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.