In this paper, we solve a long-standing open question (see, e.g., Downey [6, §7] and Downey and Moses [11]), about the spectrum of the successivity relation on a computable linear ordering. We show that if a computable linear ordering L has infinitely many successivities, then the spectrum of the successivity relation is closed upwards in the computably enumerable Turing degrees. To do this, we...

The d.c.e. degrees were first studied by Cooper [5] and Lachlan who showed that there is a proper d.c.e degree, a d.c.e. degree containing no c.e. sets, and that every nonzero d.c.e. degree bounds a nonzero c.e. degree, respectively. The main motivation of research on the d.c.e. degrees is to study the differences between the structures of d.c.e. degrees and Δ2 degrees, and between the structur...

We introduce a Π1-uniformization principle and establish its equivalence with the set-theoretic hypothesis (ω1) = ω1. This principle is then applied to derive the equivalence, to suitable set-theoretic hypotheses, of the existence of Π1 maximal chains and thin maximal antichains in the Turing degrees. We also use the Π1-uniformization principle to study Martin’s conjecture on cones of Turing de...

A class of recursively enumerable sets may be classified either as an object in itself — the range of a two-place function in the obvious way — or by means of the corresponding set of indices. The latter approach is not only more precise but also, as we show below, provides an alternative method for solving certain problems on recursively enumerable sets and their degrees of unsolvability. The ...

We show that the structure of the enumeration degrees De has a finite automorphism base consisting of finitely many total elements below the first jump of its least element. As a consequence we obtain that the rigidity of the structure of the enumeration degrees is implied by the rigidity of the local structures of the Σ2 enumeration degrees, the ∆ 0 2 Turing degrees and the computably enumerab...

Given two incomparable c.e. Turing degrees a and b, we show that there exists a c.e. degree c such that c = (a∪ c)∩ (b∪ c), a∪ c ∣∣b∪ c, and c < a ∪ b.

We investigate the connections between the complexity of a c.e. set, as calibrated by its strength as an oracle for Turing computations of functions in the Ershov hierarchy, and how strong reducibilities allow us to compute such sets. For example, we prove that a c.e. degree is totally ω-c.e. iff every set in it is weak truth-table reducible to a hypersimple, or ranked, set. We also show that a...

Suppose that R is a countable relation on the Turing degrees. Then R can be defined in D, the Turing degrees with ≤T , by a first order formula with finitely many parameters. The parameters are built by means of a notion of forcing in which the conditions are essentially finite. The conditions in the forcing partial specify finite initial segments of the generic reals and impose a infinite cons...

When bounds on complexity of some aspect of a structure are preserved under isomorphism, we refer to them as intrinsic. Here, building on work of Soskov [33], [34], we give syntactical conditions necessary and sufficient for a relation to be intrinsically Π1 on a structure. We consider some examples of computable structures A and intrinsically Π1 relations R. We also consider a general family o...

It is shown that for any computably enumerable degree a 6= 0, any degree c 6= 0, and any Turing degree s, if s ≥ 0, and c.e. in a, then there exists a c.e. degree x with the following properties, (1) x < a, c 6≤ x, (2) a is splittable over x, and (3) x = s. This implies that the Sacks’ splitting theorem and the Sacks’ jump theorem can be uniformly combined. A corollary is that there is no atomi...