The question: "What is the size of a boldface pointclass?" is trivial under the Axiom of Choice, but becomes very interesting if we restrict ourselves to the realm of definable sets and maps. Assuming the Axiom of Determinacy a complete description of the cardinality of the boldface pointclasses is given, showing that the size of a pointclass is tightly related to its descriptive-set-theoretic comlexity.
This is joint work with G. Hjorth and I. Neeman.