Abstract

For a natural number n, an n-recursive enumerable (n-r.e.) set can be defined as the symmetric difference of n recursively enumerable sets. In a series of ground-breaking papers around 1970, Ershov generalized this notion to transfinite levels based on Kleenes notations of ordinals. The corresponding Turing degree structures form a natural hierarchy below the halting problem, which is often referred as Ershov hierarchies. In this talk, I will give a survey on the early results by Ershov, and present the elementary differences between those degree structures, in the end I will mention some topics that we are working on.



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