We consider so‐called extremal numberings that form the greatest or minimal degrees under the reducibility of all A‐computable numberings of a given family of subsets of, where A is an arbitrary oracle. Such numberings are very common in the literature and they are called universal and minimal A‐computable numberings, respectively. The main question of this paper is when a universal or a minimal A‐computable numbering satisfies the Recursion Theorem (with parameters). First we prove that the Turing degree of a set (...) A is hyperimmune if and only if every universal A‐computable numbering satisfies the Recursion Theorem. Next we prove that any universal A‐computable numbering satisfies the Recursion Theorem with parameters if A computes a non‐computable c.e. set. We also consider the property of precompleteness of universal numberings, which in turn is closely related to the Recursion Theorem. Ershov proved that a numbering is precomplete if and only if it satisfies the Recursion Theorem with parameters for partial computable functions. In this paper, we show that for a given A‐computable numbering, in the general case, the Recursion Theorem with parameters for total computable functions is not equivalent to the precompleteness of the numbering, even if it is universal. Finally we prove that if A is high, then any infinite A‐computable family has a minimal A‐computable numbering that satisfies the Recursion Theorem. (shrink)

It is proved that for every countable structure and a computable successor ordinal α there is a countable structure which is ‐least among all countable structures such that is Σ‐definable in the αth jump. We also show that this result does not hold for the limit ordinal. Moreover, we prove that there is no countable structure with the degree spectrum for.

In this work, we study computable families of Turing degrees introduced and first studied by Arslanov and their numberings. We show that there exist finite families of Turing c.e. degrees both those with and without computable principal numberings and that every computable principal numbering of a family of Turing degrees is complete with respect to any element of the family. We also show that every computable family of Turing degrees has a complete with respect to each of its elements computable (...) numbering even if it has no principal numberings. It follows from results by Mal’tsev and Ershov that complete numberings have nice programming tools and computational properties such as Kleene’s recursion theorems, Rice’s theorem, Visser’s ADN theorem, etc. Thus, every computable family of Turing degrees has a computable numbering with these properties. Finally, we prove that the Rogers semilattice of each such non-empty non-singleton family is infinite and is not a lattice. (shrink)

We extend the limitwise monotonicity notion to the case of arbitrary computable linear ordering to get a set which is limitwise monotonic precisely in the non‐computable degrees. Also we get a series of connected non‐uniformity results to obtain new examples of non‐uniformly equivalent families of computable sets with the same enumeration degree spectrum.

In this paper, we prove several results about the Turing jumps of low c.e. sets. We show that only Δ‐levels of the Ershov Hierarchy can properly contain the Turing jumps of c.e. sets and that there exists an arbitrarily large computable ordinal with a normal notation such that the corresponding Δ‐level is proper for the Turing jump of some c.e. set. Next, we generalize the notion of jump traceability to the jump traceability with ‐ and ‐bound for every infinite computable (...) ordinal α. It is known that jump traceability and superlowness coincide on the c.e. sets and we show that for every infinite computable ordinal α, jump traceability with ‐ or ‐bound of a c.e. set A is equivalent to the fact that. Finally, we consider the generalized truth‐table reducibilities and prove that for every (not necessarily the Turing jump of a c.e. set) set A and every limit computable ordinal α, iff. (shrink)

The main question of this article is whether there is a family closed under finite differences (i.e., if A belongs to the family and B=∗A, then B also belongs to the family) that can be enumerated by any noncomputable c.e. degree, but which cannot be enumerated computably. This question was formulated by Greenberg et al. (2020) in their recent work in which families that are closed under finite differences, close to the Slaman–Wehner families, are deeply studied.

We introduce a hierarchy of sets which can be derived from the integers using countable collections. Such families can be coded into countable algebraic structures preserving their algorithmic properties. We prove that for different finite levels of the hierarchy the corresponding algebraic structures have different classes of possible degree spectra.