Orbit cardinalities

Abstract

This paper considers "definable cardinalities" arising from Polish group actions. The first part of the paper answers a question of Becker-Kechris by showing that under suitable determinacy assumptions in ZF+DC, every action by a Polish group on a metric space is either as complicated as the Vitali equivalence relation or no more complicated than the equality relation on bounded subsets of \aleph_1. The same argument shows that if hereditarily wellorderabel sets can be used as complete invariants, then elements in HC will do. The second half gives a new proof of Howard Becker's recent theorem that Vaught's conjecture holds for groups with a left invariant complete metric. The proof also serves to show that Vaught's conjecture holds for such groups on Σ^1_1 sets (in ZFC) and "reasonably definable" sets under large cardinal assumptions.