Abstract
A real number x is called Δ 2 0 if its binary expansion corresponds to a Δ 2 0 -set of natural numbers. Such reals are just the limits of computable sequences of rational numbers and hence also called computably approximable. Depending on how fast the sequences converge, Δ 2 0 -reals have different levels of effectiveness. This leads to various hierarchies of Δ 2 0 reals. In this survey paper we summarize several recent developments related to such kind of hierarchies shown by the author and his collaborators.
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