Numberings and randomness

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 ${\Pi }_1^0$ classes of positive measure. On the other hand, the ${\Pi }_1^0$ 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 ${\Pi }_1^0$ classes contained in the class of reals satisfying a fixed randomness constant, we prove that at least an effective numbering exists.

The final publication is available at Springer via http://dx.doi.org/10.1007/978-3-642-03073-4_6