The high/low hierarchy in the local structure of the ωω-enumeration degrees

This paper gives two definability results in the local theory of the ωω-enumeration degrees. First, we prove that the local structure of the enumeration degrees is first order definable as a substructure of the ωω-enumeration degrees. Our second result is the definability of the classes HnHn and LnLn of the highnn and lown ωω-enumeration degrees. This allows us to deduce that the first order theory of true arithmetic is interpretable in the local theory of the ωω-enumeration degrees.

► We prove that each nonzero Δ20 e-degree is cupped by a member of a KK-pair, splitting 0e′. ► We characterize the KK-pairs in the local structure of the ωω-enumeration degrees, GωGω. ► We prove that for every nn, the degree on is first order definable in GωGω. ► We prove that an isomorphic copy of the Σ20 e-degrees is definable in GωGω. ► We prove that every level of the high/low jump hierarchy is definable in GωGω.