This aspect is very interesting:
Qea. If this were normal science, the proof that P is inconsistent could be written up rather quickly. But since this work calls for a paradigm shift in mathematics, it is essential that all details be developed fully. At present, I have written just over 100 pages beginning this. The current version is posted as a work in progress at http://www.math.princeton.edu/~nelson/books.html, and the book will be updated from time to time. The proofs are automatically checked by a program I devised called qea (for quod est absurdum, since all the proofs are indirect). Most proof checkers require one to trust that the program is correct, something that is notoriously difficult to verify. But qea, from a very concise input, prints out full proofs that a mathematician can quickly check simply by inspection. To date there are 733 axioms, definitions, and theorems, and qea checked the work in 93 seconds of user time, writing to files 23 megabytes of full proofs that are available from hyperlinks in the book.
It seems easier to me to simply trust Coq than to read through 23 megabytes of proofs by eye. But I'm not entirely certain what the purpose of qea is.
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.