Numberings and Randomness

Paul Brodhead, Bjørn Kjos-Hanssen

Abstract: We prove various results on effective numberings and Friedberg numberings of families related to algorithmic randomness. The family of all Martin-Löf random left-computably enumerable reals has a Friedberg numbering, as does the family of all \(\Pi^0_1\) classes of positive measure. On the other hand, the \(\Pi^0_1\) classes contained in the Martin-Löf random reals do not even have an effective numbering, nor do the left-c.e. reals satisfying a fixed randomness constant. For \(\Pi^0_1\) classes contained in the class of reals satisfying a fixed randomness constant, we prove that at least an effective numbering exists.