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17th "Summer" Conference on Topology and Applications
July 1-4, 2002
University of Auckland
Auckland, New Zealand

Organizers
David Gauld (University of Auckland), Sina Greenwood (University of Auckland), David McIntyre (University of Auckland), Warren Moors (Waikato University), Sidney Morris (University of South Australia), Vladimir Pestov (Victoria University Wellington), Ivan Reilly (University of Auckland), Des Robbie (University of Melbourne)

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The Vaught Conjecture: a counterexample
by
R. W. Knight
University of Oxford

The Topological Vaught Conjecture is the statement that any continuous action of a Polish (sc. separable, completely metrisable) topological group on a Polish space, has countably many or 2\aleph0 orbits. The conjecture was made by D. E. Miller in 1980. It implies the Vaught Conjecture in Mathematical Logic: that any theory in a countable first order language has countably many, or 2\aleph0, countable models.

We will discuss the links between the two problems, and briefly describe a counterexample to the Vaught Conjecture, which yields a continous action of Sym(\omega), the group of permutations of the natural numbers, on a G\delta subset of the reals, with \aleph1 orbits.

Date received: May 30, 2002

Copyright © 2002 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cait-25.