Rank and Randomness

Abstract

We show that for each computable ordinal α > 0 it is possible to find in each Martin-Löf random ∆ 0 2 degree a sequence R of Cantor-Bendixson rank α, while ensuring that the sequences that inductively witness R's rank are all Martin-Löf random with respect to a single countably supported and computable measure. This is a strengthening for random degrees of a recent result of Downey, Wu, and Yang, and can be understood as a randomized version of it.