Posts programme for the Ershov hierarchy

Abstract

This article extends Posts; programme to finite levels of the Ershov hierarchy of Δ2 sets. Our initial characterization, in the spirit of Post (1994, Bulletin of the American Mathematical Society, 50, 284316), of the degrees of the immune and hyperimmune n-enumerable sets leads to a number of results setting other immunity properties in the context of the Turing and wtt-degrees derived from the Ershov hierarchy. For instance, we show that any n-enumerable hyperhyperimmune set must be co-enumerable, for each n < 2. The situation with regard to the wtt-degrees is particularly interesting, as demonstrated by a range of results concerning the wtt-predecessors of hypersimple sets.Finally, we give a number of results directed at characterizing basic classes of n-enumerable degrees in terms of natural information content. For example, a 2-enumerable degree contains a 2-enumerable dense immune set iff it contains a 2-enumerable r-cohesive set iff it bounds a high enumerable set. This result is extended to a characterization of n-enumerable degrees which bound high enumerable degrees. Furthermore, a characterization for n-enumerable degrees bounding only low2 enumerable degrees is given.