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 another way, whatever the answer ends up being, this conversation will have been wasted time in retrospect).
Sorry if I gave you a bad impression: I am not trying to start a debate in any adversarial sense. I am just curious.
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.
Of that there's no doubt, but it speaks well of Nelson that he's apparently resisted the temptation toward self-deceipt for decades, openly working on this problem the whole time.
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?
The announcement came as a surprise, so the update wasn't negligible. I probably wouldn't have gone as low as million-to-one before, but I might have been prepared to estimate a 99.9% chance that arithmetic is consistent. However, I'm not quite sure how much of this change is a Bayesian update and how much is the fact that I got a shock and thought about the issue a lot more carefully.
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.