Enumeration 1-Genericity in the Local Enumeration Degrees

Enumeration 1-genericity in the Local Enumeration Degrees

We discuss a notion of forcing that characterises enumeration 1genericity and we investigate the immunity, lowness and quasiminimality properties of enumeration 1-generic sets and their degrees. We construct an enumeration operator ∆ such that, for any A, the set ∆ is enumeration 1-generic and has the same jump complexity as A. We also prove that every nonzero ∆ 2 degree bounds a nonzero enumer...

Genericity and Non-bounding in the Enumeration degrees

The structure of the semi lattice of enumeration degrees has been investigated from many aspects. One aspect is the bounding and nonbounding properties of generic degrees. Copestake proved that every 2-generic enumeration degree bounds a minimal pair and conjectured that there exists a 1-generic degree that does not bound a minimal pair. In this paper we verify this longstanding conjecture by c...

Definability in the enumeration degrees

We prove that every countable relation on the enumeration degrees, E, is uniformly definable from parameters in E. Consequently, the first order theory of E is recursively isomorphic to the second order theory of arithmetic. By an effective version of coding lemma, we show that the first order theory of the enumeration degrees of the Σ2 sets is not decidable.

Definability in the local theory of the ω-enumeration degrees

We continue the study of the local theory of the structure of the ω-enumeration degrees, started by Soskov and Ganchev [7]. We show that the classes of 1-high and 1-low ω-enumeration degrees are definable. We prove that a standard model of arithmetic is definable as well. 1 ω-enumeration degrees In computability theory there is a vast variety of reducibility relations. Informally, the common fe...

Bounding nonsplitting enumeration degrees