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.



Has Vaught's Conjecture Been Solved?
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.


