  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
\Sigma^1_1 sets (in ZFC) and "reasonably definable" sets under large cardinal
assumptions.
