On Lusin's restricted continuum problem (Q1058511)

The author defines an equivalence relation E on the real line to be Lusin if all equivalence classes are Borel sets of bounded rank and E does not admit a non-empty perfect set of pairwise inequivalent elements. He defines a partition of the line into non-empty subsets to be Lusin if the corresponding equivalence relation is Lusin. A Lusin partition is countable if it admits at most countable many members. The author proves that in Solovay's model (in which every set of reals is Lebesgue measurable) any Lusin partition is countable; and that if for all \(\alpha\), \(\aleph_ 1^{L(\alpha)}<\aleph_ 1\), then any Lusin partition, whose corresponding equivalence relation is \(\Delta^ 1_ 2\), is countable.