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

Abstract

We prove that a sequence of sets containing representatives of cupping partners for every nonzero \({\Delta^0_2}\) enumeration degree cannot have a \({\Delta^0_2}\) enumeration. We also prove that no subclass of the \({\Sigma^0_2}\) enumeration degrees containing the nonzero 3-c.e. enumeration degrees can be cupped to \({\mathbf{0}_e'}\) by a single incomplete \({\Sigma^0_2}\) enumeration degree.