Assuming the continuum hypothesis there is an inseparable sequence of length ω1 that contains no Lusin subsequence, while if Martin’s Axiom and ¬CH is assumed then every inseparable sequence (of length ω1) is a union of countably many Lusin subsequences.

Smullyan's notion of eeectively inseparable pairs of sets is not the best eeec-tive/constructive analog of Kleene's notion of pairs of sets inseparable by a recursive set. We present a corrected notion of eeectively inseparable pairs of sets, prove a characterization of our notion, and show that the pairs of index sets eeectively inseparable in Smullyan's sense are the same as those eeectively ...