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Today I found myself at the Wikipedia page on Vaught's Conjecture,

http://en.wikipedia.org/wiki/Vaught_conjecture

and it says that Prof. Knight, of Oxford, "has announced a counterexample" to the conjecture. The phrasing is odd; I interpret it as suggesting that there is some doubt as to whether Prof. Knight attained his goal. Another Wikipedia page uses similar language, "it is thought that there is a counterexample" or something like that. (I forget which page now.)

Prof. Knight's page, which you can easily find through the link above, certainly gives the impression that he himself harbors no doubts about his achievement. Since the counterexample is a 117-page construction and not immediately perspicuous, I thought I'd ask here what the situation is. Is the paper being refereed? Apparently it's from 2002, so something unusual must be going on.

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The "it is thought that" page appears to be en.wikipedia.org/wiki/Robert_Lawson_Vaught –  David Eppstein Apr 22 '10 at 5:16
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3 Answers

up vote 18 down vote accepted

As far as I understand, no, Vaught's Conjecture has not been resolved.

We held a reading seminar on Robin Knight's proposed counter-example closely following each of his drafts and simplified presentations here at Berkeley some years ago and were ultimately convinced that the draft of January 2003 does not contain a correct disproof of Vaught's Conjecture and requires more than minor emendations to produce a complete proof. We did not discover any essential error, though there were important points in the argument where it seemed to us that even the author had not worked out the technical details.

That said, it is possible that revisions he has posted since then are sufficient, though in view of how much time it would require to enter into the details of the argument, I am not willing to work through the later papers until the basic architecture of the proof is certified by some other expert.

To be fair to Robin Knight, his work in set theoretic topology is well-respected and his construction takes into account the relevant features required for a counter-example to Vaught's Conjecture. If you would like to know whether or not he believes that his proof works, you should ask him directly. If he says that he does believe the proof to be valid, then you can attempt to check the proof yourself. The difficulty in reading his manuscript is not the amount of background material one must know in order to follow it, but exactly the opposite: almost everything is developed from scratch so that one must hold the entire construction in one's mind without having the usual anchors of established theorems.

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Thanks Thomas, that's very helpful information. I've only asked two questions so far on MO, and you've answered both! –  Pietro KC Apr 22 '10 at 5:44
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Hardly an answer, but note that Knight's webpage provides some further information (historical and otherwise):

http://people.maths.ox.ac.uk/knight/stuff/preprints.html

By coincidence, I had dinner with Nate Ackerman at a model theory conference at MIT [one of two such conferences I have ever attended, hence the coincidence] in early 2003, soon after he had spent a long time looking over Knight's proof. At the time, he was of the opinion that it was essentially correct.

My impression is that this is a long, intricate and difficult argument, so much so that most experts of the model theory community have not been able to work their way through it in detail. This is certainly not without precedent. It took the community about four years to check Perelman's arguments, and certainly there were more mathematicians qualified and highly motivated to do so than in this case. Also the Annals of Mathematics referees famously gave up on Hales' proof of the Kepler Conjecture. In that case it seems like the person who has put in the most effort to understand, simplify and popularize Hales' proof is...Hales. So maybe something similar is happening here?

Those are just outsider remarks. I look forward to hearing the real story.

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Thanks for your input, Pete! The page you link is precisely what I had in mind when I said Prof. Knight had no doubts himself. Maybe you're right. The model theory community isn't as big as the relevant geometry community in the case of Perelman, and maybe everyone is waiting for someone else to take the first step. –  Pietro KC Apr 22 '10 at 5:41
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Knight gave a serious of talks at the logic seminar at Oxford a few years ago; if I recall correctly, the seminar's conclusion was similar to what's described by Thomas Scanlon above.

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