We give an alternative definition of the enumeration jump operator. We prove that the class of total enumeration degrees and the class of low enumeration degrees are first order definable in theâ€¦ (More)

We say that A â‰¤LR B if every B-random number isA-random. Intuitively this means that if oracle A can identify some patterns on some real !, oracle B can also find patterns on !. In other words, B isâ€¦ (More)

This paper gives two definability results in the local theory of the Ï‰-enumeration degrees. First we prove that the local structure of the enumeration degrees is first order definable as aâ€¦ (More)

We prove that a sequence of sets containing representatives of cupping partners for every nonzero âˆ†2 enumeration degree cannot have a âˆ† 0 2 enumeration. We also prove that no subclass of the Î£ 2â€¦ (More)

We prove that the cototal enumeration degrees are exactly the enumeration degrees of sets with good approximations, as introduced by Lachlan and Shore [17]. Good approximations have been used as aâ€¦ (More)

The continuous degrees measure the computability-theoretic content of elements of computable metric spaces. They properly extend the Turing degrees and naturally embed into the enumeration degrees.â€¦ (More)

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. Weâ€¦ (More)

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

We study Kalimullin pairs, a definable class (of pairs) of enumeration degrees that has been used to give first-order definitions of other important classes and relations, including the enumerationâ€¦ (More)