Nelson's program to show inconsistency of ZF 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?
 A: I cannot judge how serious these are, I just put  Nelson predicative arithmetic in Google and came up with lots of stuff:
Link
"at the Nelson meeting in Vancouver in June 2004."
http://citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.110.478
"This paper starts by discussing Nelson’s philosophy of mathematics, which is a blend of mathematical formalism and a radical constructivism. As such, it makes strong assertions about the foundations of mathematic and the reality of mathematical objects."
http://math.ucsd.edu/~sbuss/ResearchWeb/nelson/
http://www.illc.uva.nl/Publications/ResearchReports/X-1989-01.text.pdf
This one is skeptical:
Link
A: 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
A: 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.
A: This is perhaps an obvious remark, but it may be helpful for those who haven't yet gotten used to the fact that one must think about consistency questions slightly differently from how we think of "ordinary" mathematical questions.  Namely, let us ask what an "obstruction to finding an inconsistency in ZF" might look like?  The obvious "obstruction" would be a proof that ZF is consistent.  But we can't expect to find such a thing, by Goedel's 2nd incompleteness theorem.  Therefore, we cannot hope to find a mathematical obstruction in the usual sense.
