Preprints

Vaught's conjecture

Vaught's conjecture is the statement that any theory in a countable language of first-order 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 proof sections of the paper.

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. Parts one and two have had minor corrections made following a communication by Nate Ackerman. 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. The third part shows how a structure as described in part two can be constructed from what I refer to as a Vaughtian Stack; these have a formidable definition but the idea is that they should be relatively easy to build. I will post the fourth part as opportunity permits (the academic year between October and March is, as you can imagine, extremely busy).

Other topics

Non-hyperbolic one-dimensional invariant sets with a countably infinite collection of inhomogeneities, co-authored with Chris Good and Brian Raines provides examples of solenoid-like spaces (specifically, inverse limits of intervals under tent-maps) of uncountably many homeomorphism classes, using countable well-founded trees.

Characterizing continuous functions on compact spaces, co-authored 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, co-authored with P. M. Gartside, is in volume 35 of the Bulletin of the London Mathematical Society (2003), pages 624-634. 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.