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Turing degrees and the Ershov hierarchy
 in Proceedings of the Tenth Asian Logic Conference, Kobe, Japan, 16 September 2008, World Scienti…c
"... Abstract. An nr.e. set can be defined as the symmetric difference of n recursively enumerable sets. The classes of these sets form a natural hierarchy which became a wellstudied topic in recursion theory. In a series of groundbreaking papers, Ershov generalized this hierarchy to transfinite level ..."
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Abstract. An nr.e. set can be defined as the symmetric difference of n recursively enumerable sets. The classes of these sets form a natural hierarchy which became a wellstudied topic in recursion theory. In a series of groundbreaking papers, Ershov generalized this hierarchy to transfinite levels based on Kleene’s notations of ordinals and this work lead to a fruitful study of these sets and their manyone and Turing degrees. The Ershov hierarchy is a natural measure of complexity of the sets below the halting problem. In this paper, we survey the early work by Ershov and others on this hierarchy and present the most fundamental results. We also provide some pointers to concurrent work in the field. 1.