At the end of the paper Division by three by Peter G. Doyle and John H. Conway, the authors say:
Not that we believe there really are any such things as infinite sets, or that the Zermelo-Fraenkel axioms for set theory are necessarily even consistent. Indeed, we’re somewhat doubtful whether large natural numbers (like $80^{5000}$, or even $2^{200}$) exist in any very real sense, and we’re secretly hoping that Nelson will succeed in his program for proving that the usual axioms of arithmetic—and hence also of set theory—are inconsistent. (See Nelson [E. Nelson. Predicative Arithmetic. Princeton University Press, Princeton, 1986.].) All the more reason, then, for us to stick with methods which, because of their concrete, combinatorial nature, are likely to survive the possible collapse of set theory as we know it today.
Here are my questions:
What is the status of Nelson's program? Are there any obstruction to finding a relatively easy proof of the inconsistency of ZF? Is there anybody seriously working on this?
Best Answer
Nelson claimed to have succeeded just now.
http://www.math.princeton.edu/~nelson/papers/outline.pdf
I hope consensus about this forms soon, so I can know what to do with the rest of my life. If only I had been born a few years later, I wouldn't be put into the position of worrying that my chosen career path is doomed and I must go build houses or something.
Update:
As per Michael's comment, the claim has been withdrawn.