has your prior for "a contradiction in PA will be found within a hundred years" moved since Nelson's announcement?
Yes, obviously P(respectable mathematician claims a contradiction | a contradiction exists) > P(respectable mathematician claims a contradiction | no contradiction exists), so it has definitely moved my estimate.
Like yours, it also moved back down when Tao responded, back up a bit when Nelson responded to him, and back down a bit more when Tao responded to him and I finally managed a coherent guess at what they were talking about.
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.
I'm not sure this is an important difference. I think scepticism about P =! NP proofs might well be just a valid even if far fewer people were working on it. If anything it would be more valid, lots of failed proofs gives you lots of chances to learn from the mistakes of others, as well as building avoiding routes which are proven not to work by others in the field. Furthermore, the fact that huge numbers of mathematicians work on P vs NP but have never claimed a proof suggests a selection effect in favour of those who do claim proofs, which is absent in the case of inconsistency.
Furthermore, not wanting to be unfair to Nelson, but the fact he's working alone on a task most mathematicians consider a waste of time may suggest a substantial ideological axe to grind (what I have heard of him supports this thoery) and sadly it is easier to come up with a fallacious proof for something when you want it to be true.
I'm not sure if this line of debate is a productive one, the issue will be resolved one way or the other by actual mathematicians doing actual maths, not by you and me debating about priors (to put it another way, whatever the answer ends up being, this conversation will have been wasted time in retrospect).
Yes, obviously P(respectable mathematician claims a contradiction | a contradiction exists) > P(respectable mathematician claims a contradiction | no contradiction exists), so it has definitely moved my estimate.
Can you roughly quantify it? Are we talking from million-to-one to million-to-one-point-five, or from million-to-one to hundred-to-one?
...I'm not sure if this line of debate is a productive one, the issue will be resolved one way or the other by actual mathematicians doing actual maths, not by you and me debating about priors (to put it anot
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.