The issue is mainly that I don't think I'm qualified to determine the outcome of the debate, Nelson and Tao are both vastly better at maths than me, so I don't have much to go on. I suspect others are in a similar predicament.
I've been trying to understand their conversation and if I understand correctly Tao is right and Nelson has made a subtle and quite understandable error, but I also estimate P(my understanding is correct) < 0.5 is not very large, so this doesn't help much, and even if I am right there could easily be something I'm missing.
I would also assign this theorem quite a low prior. Compare it to P =! NP, when a claimed proof of that comes out, mathematicians are usually highly sceptical (and usually right), even if the author is a serious mathematician like Deolalikar rather than a crank. Another example that springs to mind is Cauchy's failed proof of Fermat's last theorem (even though that was eventually proven, I still think a low prior was justified in both cases).
This, if correct, would be a vastly bigger result than either of those. I don't think it would be an exaggeration to call this the single most important theorem in the history of mathematics if correct, so I think it deserves a much lower prior than them. Even more so, since in the case of P =! NP most mathematicians at least think it's true, even if they are sceptical of most proofs, in this case most mathematicians would probably be happy to bet against it (that counts for something, even if you disagree with them).
I don't have much money to lose at this point in my life, but I'd be happy to bet $50 and $1 that this is wrong.
I think there's a salient difference between this and P = NP or other famous open problems. P = NP is something that thousands of people are working on and have worked on over decades, while "PA is inconsistent" is a much lonelier affair. A standard reply is that every time a mathematician proves an interesting theorem without encountering a contradiction in PA, he has given evidence for the consistency of PA. For various reasons I don't see it that way.
Same question as for JoshuaZ: has your prior for "a contradiction in PA will be found within a hundred years" moved since Nelson's announcement?
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.