On many-one degrees

Abstract

Two questions were left open in [1]. 1) Do W(σ) and W(r) have the same many-one degree for any σ, τ such that ωη+1.2 ≤ σ < τ < ωη+2? 2) Is W(σ) many-one reducible to W(r) for any σ, τ such that ωη+1.2 ≤ σ < τ < ωη+2? We answered the second question affirmatively in [2]. In this paper we answer the first question in the negative. From the results of [1] [2] and this paper we have a complete picture of the many-one degrees of the predicates e ∈ W(σ)