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gwern comments on Edward Nelson claims proof of inconsistency in Peano Arithmetic - Less Wrong
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Nelson has withdrawn his claim. Link.
Is that a withdrawal of the entire approach or just one part of it?
On the FOM list, he writes:
Thanks, that's very useful for http://predictionbook.com/predictions/3508
I am interested in how this event has influenced peoples view of the consistency of arithmetic. For those who had a viewpoint on it prior to this, what would you have estimated before and after and with what confidence? I will not state my own view for fear of damaging the results.
It seems important in this context to distinguish between consistency of different systems
Overall, my consistency estimate for PA if anything has gone up in this context. Because if Edward Nelson tries really hard and fails to find a contradiction that's more evidence that there isn't one. How much should it go up by? I'm not sure.
On the other hand, this may very well make my estimate for consistency of ZFC go down. That's because although in this specific case Nelson's approach didn't work, it opens up new avenues into demonstrating inconsistency, and part of me could see this sort of thing being more successfully applied to ZFC (although that also seems unlikely). (Also note that there have been attempts by a much larger set of mathematicians in the last few years to find an inconsistency in ZF and that has essentially failed.)
One thing that this has also done is made me very aware in a visceral fashion that number theory and logic are not the same things even when one is talking about subsystems of PA. I already knew that, but maybe had not fully emotionally processed it as much as I should have as when I saw Nelson's outline, and then the subsequent discussions and had a lot of trouble following the details. The main impact of this is to suggest that when making logic related claims (especially consistency and independence/undecidability issues) I should probably not rate my own expertise as highly as I do. As a working mathematician, I almost certainly have more relevant expertise than a random individual, but I've probably been overestimating how much my expertise matters. In both the cases of ZF/ZFC and the case of PA, reducing my confidence means increasing the chance that they are inconsistent. But, I'm not at all sure by how much this should matter. So maybe this should leave everything alone?
So overall I'd say around .995 consistency for PA and .99 consistency for ZF. Not much change from the old values.