The jump hierarchy in the enumeration degrees

Abstract

We show that all levels of the jump hierarchy are first order definable in the local structure of the enumeration degrees.

A Structural Dichotomy in the Enumeration Degrees

“…He then used them to define the enumeration jump operator. Ganchev and Soskova [6] proved that -pairs are definable in , the substructure of the enumeration degrees, and used them to prove the first order definability of a series of subclasses of , including the total enumeration degrees [7], the downwards properly enumeration degrees, the upwards properly enumeration degrees [6], and the Low and High enumeration degrees, for all [8]. A special type of -pairs—the maximal -pairs—were used by Cai et al [3] to define the total enumeration degrees.…”