Definable cardinals just beyond R/Q

We establish the inexistence of countable bases for the family of definable cardinals associated with countable Borel equivalence relations which are not measure reducible to E₀, thereby ruling out natural generalizations of the Glimm-Effros dichotomy. We also push the primary known results concerning the abstract structure of the Borel cardinal hierarchy nearly to its base, using arguments substantially simpler than those previously employed. Our main tool is a strong notion of separability, which holds of orbit equivalence relations induced by group actions satisfying an appropriate measureless local rigidity property.