On ∑11 equivalence relations with borel classes of bounded rank

Journal of Symbolic Logic 49 (4):1273 - 1283 (1984)
In Baire space N = ω ω we define a sequence of equivalence relations $\langle E_\nu| \nu , each E v being Σ 1 1 with classes in Π 0 1 + ν + 1 and such that (i) E ν does not have perfectly many classes, and (ii) N/E ν is countable iff $\omega^L_\nu . This construction can be extended cofinally in (δ 1 3 ) L . A new proof is given of a theorem of Hausdorff on partitions of R into ω 1 many Π 0 3 sets
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