Model Theory and the Vaught Conjecture

Abstract

The Vaught Conjecture is a notorious open problem in mathematical logic. A number of stronger conjectures have subsequently been proposed, some of which are conjectures about the descriptive set theory of Polish group actions. These later conjectures are known (collectively) as the Topological Vaught Conjecture. In §6.1 we give some background on the Vaught Conjecture. This section is written primarily for non-logicians, and contains nothing that is not well known to specialists. §6.2 contains a number of results, some new, about the Topological Vaught Conjecture. Our new results are of two types: First, that various conjectures imply - or are equivalent to - various other conjectures, and, second, that certain even stronger conjectures are false. There are few things to be proved in §6.2; the results here are mostly corollaries of theorems in earlier parts of this book. In §6.3 we prove a theorem about atomic models (which has nothing to do with the Vaught Conjecture). More connections between group actions and logic will be discussed in §7.