Effective Strong Nullness and Effectively Closed Sets

Abstract

The strongly null sets of reals have been widely studied in the context of set theory of the real line. We introduce an effectivization of strong nullness. A set of reals is said to be effectively strongly null if, for any computable sequence {ε _n } _n ∈ ω of positive rationals, a sequence of intervals I _n of diameter ε _n covers the set. We show that for $\Pi^0_1$ subsets of 2^ω effective strong nullness is equivalent to another well studied notion called diminutiveness: the property of not having a computably perfect subset. In addition, we also investigate the Muchnik degrees of effectively strongly null $\Pi^0_1$ subsets of 2^ω. Let MLR and DNC be the sets of all Martin-Löf random reals and diagonally noncomputable functions, respectively. We prove that neither the Muchnik degree of MLR nor that of DNC is comparable with the Muchnik degree of a nonempty effectively strongly null $\Pi^0_1$ subsets of 2^ω with no computable element.