On the jump classes of noncuppable enumeration degrees

Abstract

We prove that for every Σ⁰₂ enumeration degree b there exists a noncuppable Σ⁰₂ degree a > 0_e such that b' ≤_c a' and a'' ≤_c b''. This allows us to deduce, from results on the high/low jump hierarchy in the local Turing degrees and the jump preserving properties of the standard embedding ι : 𝒟_T → 𝒟_e, that there exist Σ⁰₂ noncuppable enumeration degrees at every possible—i.e., above low₁—level of the high/low jump hierarchy in the context of 𝒟_e.