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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?

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    $\begingroup$ Note that on the first page footnotes, Doyle says about Conway $$ $$ But he has never approved of this exposition, which he regards as full of `fluff.' $\endgroup$
    – Will Jagy
    Commented Aug 25, 2010 at 21:50
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    $\begingroup$ A central obstruction to finding an "easy" proof of the syntactic inconsistency of ZF is that nobody has managed to do it in 80+ years, and not for lack of trying. More recently, there was fear (or hope) that large cardinals might be useful for finding inconsistencies in ZF, but that idea hasn't panned out either. Arithmetic is even worse: there are multiple, unrelated consistency proofs for arithmetic, so it would be remarkable to find a syntactic inconsistency there. $\endgroup$
    – Carl Mummert
    Commented Aug 25, 2010 at 21:55
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    $\begingroup$ This is not a disagreement with what Carl Mummert says but it is worth remembering that when Zermelo first proposed his axioms for set theory, there was considerable scepticism that they really would avoid contradictions. People like Bertrand Russell, Philip Jourdain and Henri Poincaré criticised his axioms. Russell wrote that "I suspect that his axioms will not really avoid contradictions, i.e., I suspect new contradictions could be manufactured specially designed to be consistent with his axioms." [quoted on p. 91 of Ebbinghaus's biography of Zermelo tinyurl.com/2fskff7 ] $\endgroup$
    – Marko Amnell
    Commented Aug 26, 2010 at 7:50
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    $\begingroup$ math.princeton.edu/~nelson/papers.html $\endgroup$
    – Kaveh
    Commented Aug 26, 2010 at 9:20
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    $\begingroup$ Note that the question states that Nelson's program is about the consistency of arithmetic. So there is even more than 80 years of work that did not come up with inconsistencies. The Russell quote by Marko Amnell is interesting. If you change "contradictions" to "undecidable statements" you get a prediction of the incompleteness theorems. And I would guess that the incompleteness phenomenon is something that Russell wouldn't have thought of at that time. $\endgroup$
    – Stefan Geschke
    Commented Aug 26, 2010 at 10:41

4 Answers 4

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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.

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    $\begingroup$ Maybe I'm missing something but what do the above two links have to do with the topic? $\endgroup$
    – Opt
    Commented Sep 27, 2011 at 4:07
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    $\begingroup$ @Sid: I think Will is, in his idiosyncratic way (no offence meant) riffing on the last sentence of mbsq. $\endgroup$
    – Yemon Choi
    Commented Sep 27, 2011 at 4:59
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    $\begingroup$ Nelsons article links to a book he is currently writing on the topic: math.princeton.edu/~nelson/books/elem.pdf $\endgroup$
    – Michael Bächtold
    Commented Sep 27, 2011 at 11:29
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    $\begingroup$ It seems Nelson has withdrawn his claim. In any case I'd like to point out that the first section of chapter 1 of his (unfinished) book on the subject (math.princeton.edu/~nelson/books/elem.pdf, "Potential vs. actual infinity") still stands. I find it to be a great motivation for his search, regardless of the outcome. $\endgroup$
    – godelian
    Commented Oct 1, 2011 at 15:40
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    $\begingroup$ the claim was withdrawn: golem.ph.utexas.edu/category/2011/09/… $\endgroup$
    – Michael Bächtold
    Commented Oct 1, 2011 at 15:41
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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.

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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.

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    $\begingroup$ I do not think I properly understand this comment. By the 2nd IT you cannot expect to prove Con(ZFC), providing ZFC is consistent. But in case you do not believe in consistency of ZFC you are not forbidden (at least by 2nd IT) to hope to show that $ZFC\vdash Con(ZFC)$. (This is not that I do not believe in consistency of ZFC, I just do not understand the argument.) $\endgroup$
    – Rafał Gruszczyński
    Commented Jan 25, 2013 at 15:07
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    $\begingroup$ If you show that ZF⊢Con(ZF) then it immediately follows that ZF is inconsistent, by Goedel's 2nd. That is, far from finding an obstruction to finding an inconsistency in ZF, you've actually found the exact opposite, namely an inconsistency in ZF. Probably you're so used to thinking that a proof of something in ZF tells you that it's true that you are misled into thinking that showing that ZF⊢Con(ZF) would show that ZF really is consistent. But in fact ZF⊢Con(ZF) shows exactly the opposite, that ZF is really inconsistent. $\endgroup$
    – Timothy Chow
    Commented Jan 25, 2013 at 16:52
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    $\begingroup$ But this is exactly my point - if you want to prove inconsistency of some axiomatizable theory $T$ extending, say Peano Artihmetic, show $T\vdash Con(T)$ and apply Goedel's 2nd. And my point is that you cannot hope to prove such a thing ONLY IF you believe in consistency of $T$. Otherwise, why not to look for a proof of $Con(T)$ within $T$ itself to demonstrate $T$ is flawed. Actually, I do not believe that this would be the easiest way to prove $T$'s inconsistency, but at least a possible way. $\endgroup$
    – Rafał Gruszczyński
    Commented Jan 26, 2013 at 8:19
  • $\begingroup$ O.K., I think I see your confusion. When I said "a proof that ZF is consistent," I meant an ordinary mathematical proof, of the type you read in journals and find yourself persuaded by. I did not mean a formal proof of Con(ZF) from the axioms of ZF. As you say, the latter is something that a skeptic about ZF might hope to discover. But neither the skeptic nor the believer can hope to find a proof that ZF is consistent in the sense that I intended it. $\endgroup$
    – Timothy Chow
    Commented Jan 27, 2013 at 3:38
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    $\begingroup$ I mean a proof by whatever methods the skeptic accepts as establishing mathematical truths. $\endgroup$
    – Timothy Chow
    Commented Jan 28, 2013 at 15:22
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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

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