Cupping and definability in the local structure of the enumeration degrees

Abstract

We show that every splitting of 0_e' in the local structure of the enumeration degrees, 𝔊_e, contains at least one low-cuppable member. We apply this new structural property to show that the classes of all 𝓚-pairs in 𝔊_e, all downwards properly Σ⁰₂ enumeration degrees and all upwards properly Σ⁰₂ enumeration degrees are first order definable in 𝔊_e.