TITLE: Nelson's program to show inconsistency of ZF
QUESTION [68 upvotes]: 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?

REPLY [29 votes]: Edward Nelson passed away at age 82 on September 10, 2014. You can read a tribute to Nelson's illustrious career from Princeton University.
Although he worked on the inconsistency of PA until the end, there were no standing claims to a proof of the inconsistency of PA at the time of Nelson's unfortunate passing.

On September 30, 2015, Nelson's unfinished manuscripts titled Inconsistency of Primitive Recursive Arithmetic and Elements have been posted on the arXiv, with a foreword by Sarah Jones Nelson and an afterword by Sam Buss and Terry Tao.

arxiv.org/abs/1509.09209
arxiv.org/abs/1510.00369
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