An Ulm-type classification theorem for equivalence relations in Solovay model

Abstract

AbstractWe prove that in the Solovay model, every OD equivalence relation, Ε, over the reals, either admits an OD reduction to the equality relation on the set of all countable (of length < ω1) binary sequences, or continuously embeds Ε0, the Vitali equivalence.If Ε is a (resp. ) relation then the reduction above can be chosen in the class of all Δ1 (resp. Δ2) functions.The proofs are based on a topology generated by OD sets.