Turing Degrees in Refinements of the Arithmetical Hierarchy

详细

We investigate the problem of characterizing proper levels of the fine hierarchy (up to Turing equivalence). It is known that the fine hierarchy exhausts arithmetical sets and contains as a small fragment finite levels of Ershov hierarchies (relativized to ∅ⁿ, n < ω), which are known to be proper. Our main result is finding a least new (i.e., distinct from the levels of the relativized Ershov hierarchies) proper level. We also show that not all new levels are proper.