The limitations of cupping in the local structure of the enumeration degrees

摘要

We prove that a sequence of sets containing representatives of cupping partners for every nonzero Δ_2~0 enumeration degree cannot have a Δ_2~0 enumeration. We also prove that no subclass of the Σ_2~0 enumeration degrees containing the nonzero 3-c.e. enumeration degrees can be cupped to 0_e~′ by a single incomplete Σ~0_2 enumeration degree.