The high/low hierarchy in the local structure of the [formula omitted]-enumeration degrees

Abstract

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