Cupping Classes of Σ0 2 Enumeration Degrees

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

We prove that a sequence of sets containing representatives of cupping partners for every nonzero ∆2 enumeration degree cannot have a ∆ 0 2 enumeration. We also prove that no subclass of the Σ 2 enumeration degrees containing the nonzero 3-c.e. enumeration degrees can be cupped to 0e by a single incomplete Σ 2 enumeration degree.

Cupping and definability in the local structure of the enumeration degrees

We show that every splitting of 0e in the local structure of the enumeration degrees, Ge, contains at least one low-cuppable member. We apply this new structural property to show that the classes of all K-pairs in Ge, all downwards properly Σ2 enumeration degrees and all upwards properly Σ2 enumeration degrees are first order definable in Ge.

Cupping Classes of Enumeration Degrees

Cupping and noncupping in the enumeration

We prove the following three theorems on the enumeration degrees of Σ2 sets. Theorem A: There exists a nonzero noncuppable Σ 0 2 enumeration degree. Theorem B: Every nonzero ∆2 enumeration degree is cuppable to 0e by an incomplete total enumeration degree. Theorem C: There exists a nonzero low ∆2 enumeration degree with the anticupping property.

Cupping D20 Enumeration Degrees to 0 e '

In this paper we prove that every nonzero ∆2 e-degree is cuppable to 0e by a 1-generic ∆ 0 2 e-degree (so low and nontotal) and that every nonzero ω-c.e. e-degree is cuppable to 0e by an incomplete