In this paper, we prove that there is a Π01 class in 2ω with a unique nonrecursive member, with that member a cohesive set. This solves an open question from Cenzer. The proof uses the Δ03 method in the context of the construction of a Π01 class.

The main result of this paper is to show that for every recursive ordinal α ≠ 0 and for every nonrecursive r.e. degree d there is a r.e. set of rank α and degree d.

Jockusch showed that 2-generic degrees are downward dense below a 2-generic degree. That is, if a is 2-generic, and $0 < {\bf{b}} < {\bf{a}}$, then there is a 2-generic g with $0 < {\bf{g}} < {\bf{b}}.$ In the case of 1-generic degrees Kumabe, and independently Chong and Downey, constructed a minimal degree computable from a 1-generic degree. We explore the tightness of these results.We solve a question of Barmpalias and Lewis-Pye by constructing a minimal degree computable from a weakly 2-generic (...) one. While there have been full approximation constructions of ${\rm{\Delta }}_3^0$ minimal degrees before, our proof is rather novel since it is a computable full approximation construction where both the generic and the minimal degrees are ${\rm{\Delta }}_3^0 - {\rm{\Delta }}_2^0$. (shrink)

We show the existence of noncomputable oracles which are low for Demuth randomness, answering a question in [15] (also Problem 5.5.19 in [34]). We fully characterize lowness for Demuth randomness using an appropriate notion of traceability. Central to this characterization is a partial relativization of Demuth randomness, which may be more natural than the fully relativized version. We also show that an oracle is low for weak Demuth randomness if and only if it is computable.

We study the weak truth-table and truth-table degrees of the images of subsets of computable structures under isomorphisms between computable structures. In particular, we show that there is a low c.e. set that is not weak truth-table reducible to any initial segment of any scattered computable linear ordering. Countable $\Pi _{1}^{0}$ subsets of 2ω and Kolmogorov complexity play a major role in the proof.

Cenzer, D., R. Downey, C. Jockusch and R.A. Shore, Countable thin Π01 classes, Annals of Pure and Applied Logic 59 79–139. A Π01 class P {0, 1}ω is thin if every Π01 subclass of P is the intersection of P with some clopen set. Countable thin Π01 classes are constructed having arbitrary recursive Cantor- Bendixson rank. A thin Π01 class P is constructed with a unique nonisolated point A and furthermore A is of degree 0’. It is shown that no (...) set of degree ≥0” can be a member of any thin Π01 class. An r.e. degree d is constructed such that no set of degree d can be a member of any thin Π01 class. It is also shown that between any two distinct comparable r.e. degrees, there is a degree that contains a set which is of rank one in some thin Π01 class. It is shown that no maximal set can have rank one in any Π01 class, while there exist maximal sets of rank 2. The connection between Π01 classes, propositional theories and recursive Boolean algebras is explored, producing several corollaries to the results on Π01 classes. For example, call a recursive Boolean algebra thin if it has no proper nonprincipal recursive ideals. Then no thin recursive Boolean algebra can have a maximal ideal of degree ≥0”. (shrink)