Numberings and randomness

TL;DR

This paper investigates the effective numbering of families related to algorithmic randomness, establishing which classes have Friedberg numberings and which do not, with implications for understanding the structure of random reals.

Contribution

It proves new results on the existence and non-existence of effective and Friedberg numberings for various classes of algorithmically random reals and related sets.

Findings

The family of all Martin-Löf random c.e. reals has a Friedberg numbering.

The family of all positive measure $ ext{Pi}^0_1$ classes has a Friedberg numbering.

Certain subclasses, like $ ext{Pi}^0_1$ classes within random reals, lack effective numberings.

Abstract

We prove various results on effective numberings and Friedberg numberings of families related to algorithmic randomness. The family of all Martin-L\"of 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-L\"of 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.