Relative randomness and cardinality (Q985004)

The paper studies the notion of lowness for the class of Martin-Löf random sets. A set \(B\) is low for Martin-Löf random, if every Martin-Löf random set is also Martin-Löf random relative to \(B\). This notion was introduced by \textit{A. Kučera} and \textit{S. A. Terwijn} [J. Symb. Log. 64, No. 4, 1396--1402 (1999; Zbl 0954.68080)]. Here, it is shown that if \(B\) is \(\Delta^0_2\) then \(B\) is low for Martin-Löf random iff \({\mathcal C}^B\) is countable, where \({\mathcal C}^B\) is the class of those sets \(A\) with the property that every set Martin-Löf random relative to \(B\) is also Martin-Löf random relative to \(A\).