This section is concerned with two types of question: Given sets Q and Q′, do they have the same “definable cardinality”, i.e., does there exist a “definable” bijection between them, and does Q have smaller “definable cardinality” than Q′? Usually, Q = A/E and Q′= A'/E′ are quotient spaces of an equivalence relation on a separable metric space. We are particularly interested in the case where the equivalence relation is induced by an action. This topic is related to much current research in set theory, but may be alien to those whose interest in Polish group actions derives from some part of mathematics other than set theory. We give some background on the set theory in §8.1. There are many more open questions than there are theorems. Our new results are corollaries of results in previous sections of this book. In §8.1 we consider arbitrary Polish groups; in §8.2 specific groups, particularly S∞.

Orbit cardinality

It is best to consider this subject under the assumption of ADℝ (the axiom of determinacy for games on reals). This axiom contradicts AC, but as pointed out prior to Corollary 5.3.3, there are technical reasons for working with these strange axioms. Specifically, Woodin (unpublished) has proved from large cardinal axioms (which may well be true in the real world, that is, the world of AC) that ADℝ is true in certain inner models of ZF+DC containing all reals.