
PreprintsI plan to overhaul this page over the next few weeks.Vaught's conjectureVaught's conjecture is the statement that any theory in a countable language of firstorder predicate calculus has only countably many, or continuously many, distinct countable models. The first draft of a proposed counterexample (released in May 2002) is a .ps of 117 pages. The example is unstable. I have also written a greatly expanded second draft
(posted on 3 January 2003) of an
informal description of the construction.
This does not follow the preprint through in order; rather it explains
some of the ideas that went into it. New elements include an explanation
of the ordered structure Omega, of sensible trees and ambiguity trees,
and of the There is now also a second draft of the example, taking into account many suggestions and corrections that I have received. However, many valuable and insightful suggestions have not yet been acted on, and may be carried through in future drafts. There is also a file of file of emendations to the second draft. I have produced a greatly simplified example, in four parts. In the first part, a simplified version is described of the machinery of type categories, and the formalisation in it of the Morley hierarchy of type spaces, found in my paper Categories of topological spaces and scattered theories in the Notre Dame Journal of Formal Logic. In the second part, it is shown how the theory outlined in the first part can be developed, if the spaces of types are not equipped with the natural total maps corresponding to the inclusion of one tuple in another, but with partial functions instead. In the third part, the material in the first two parts is exploited to reduce Vaught's Conjecture for the most common infinitary logic to the construction of a Vaughtian Stack. In the fourth part, which will appear shortly, a Vaughtian Stack is constructed. Other topicsNonhyperbolic onedimensional invariant sets with a countably infinite collection of inhomogeneities, coauthored with Chris Good and Brian Raines provides examples of solenoidlike spaces (specifically, inverse limits of intervals under tentmaps) of uncountably many homeomorphism classes, using countable wellfounded trees. Characterizing continuous functions on compact spaces, coauthored with Chris Good, Sina Greenwood, David McIntyre and Steve Watson, provides a complete characterisation of those actions of the semigroup (N,+) on any set X for which there is a compact Hausdorff topology on X according to which the action is continuous. This is due to appear in Advances in Mathematics. Ubiquity of free subgroups, coauthored with P. M. Gartside, is in volume 35 of the Bulletin of the London Mathematical Society (2003), pages 624634. It characterises the situation when most (in the sense of category) finitely generated, or compactly generated, or countably generated, subgroups of a Polish group are free. 
This page last modified
by R. W. Knight
2nd October 2018
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knight@maths.ox.ac.uk