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

Journal of Symbolic Logic 62 (4):1333-1351 (1997)
We prove that in the Solovay model, every OD equivalence relation, E, over the reals, either admits an OD reduction to the equality relation on the set of all countable (of length $ ) binary sequences, or continuously embeds E 0 , the Vitali equivalence. If E is a Σ 1 1 (resp. Σ 1 2 ) relation then the reduction above can be chosen in the class of all ▵ 1 (resp. ▵ 2 ) functions. The proofs are based on a topology generated by OD sets
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DOI 10.2307/2275646
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Vladimir Kanovei (2000). Linearization of Definable Order Relations. Annals of Pure and Applied Logic 102 (1-2):69-100.

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