Chapter 1: Introduction. S = <{We}c<w; C,U,n,0,w> is the substructure formed by restricting the lattice <^P(w); C , U, n,0,w> to the re subsets We of the ...

We show that one can solve Post's Problem by constructing generic sets in the usual set theoretic framework applied to tiny universes. This method leads to a new class of recursively enumerable sets: r.e. generic sets. All r.e. generic sets are low and simple and therefore of Turing degree strictly between 0 and 0'. Further they supply the first example of a class of low recursively enumerable sets which are automorphic in the lattice E of recursively enumerable sets with inclusion. (...) We introduce the notion of a promptly simple set. This describes the essential feature of r.e. generic sets with respect to automorphism constructions. (shrink)

We show that the partial order of Σ0 3-sets under inclusion is elementarily definable with parameters in the semilattice of r.e. wtt-degrees. Using a result of E. Herrmann, we can deduce that this semilattice has an undecidable theory, thereby solving an open problem of P. Odifreddi.

A recursively enumerable splitting of an r.e. setAis a pair of r.e. setsBandCsuch thatA=B∪CandB∩C= ⊘. Since for such a splitting degA= degB∪ degC, r.e. splittings proved to be a quite useful notion for investigations into the structure of the r.e. degrees. Important splitting theorems, like Sacks splitting [S1], Robinson splitting [R1] and Lachlan splitting [L3], use r.e. splittings.Since each r.e. splitting of a set induces a splitting of its degree, it is natural to study the relation between the degrees of (...) r.e. splittings and the degree splittings of a set. We say a setAhas thestrong universal splitting property if each splitting of its degree is represented by an r.e. splitting of itself, i.e., if for degA=b∪cthere is an r.e. splittingB, CofAsuch that degB=band degC=c. The goal of this paper is the study of this splitting property.In the literature some weaker splitting properties have been studied as well as splitting properties which imply failure of the SUSP. (shrink)

By constructing a maximal incomplete d.r.e. degree, the nondensity of the partial order of the d.r.e. degrees is established. An easy modification yields the nondensity of the n-r.e. degrees and of the ω-r.e. degrees.

By constructing a maximal incomplete d.r.e. degree, the nondensity of the partial order of the d.r.e. degrees is established. An easy modification yields the nondensity of the n-r.e. degrees and of the ω-r.e. degrees.

We show that the existence of a recursively enumerable set whose Turing degree is neither low nor complete cannot be proven from the basic axioms of first order arithmetic (P -) together with Σ 2 -collection (BΣ 2 ). In contrast, a high (hence, not low) incomplete recursively enumerable set can be assembled by a standard application of the infinite injury priority method. Similarly, for each n, the existence of an incomplete recursively enumerable set that is neither low n nor (...) high n - 1 , while true, cannot be established in P - + BΣ n + 1 . Consequently, no bounded fragment of first order arithmetic establishes the facts that the high n and low n jump hierarchies are proper on the recursively enumerable degrees. (shrink)

A splitting A1A2 = A of an r.e. set A is called a Friedberg splitting if for any r.e. set W with W — A not r.e., W — Ai≠0 for I = 1,2. In an earlier paper, the authors investigated Friedberg splittings of maximal sets and showed that they formed an orbit with very interesting degree-theoretical properties. In the present paper we continue our investigations, this time analyzing Friedberg splittings and in particular their orbits and degrees for various classes (...) of r.e. sets. (shrink)

In this article we show that it is possible to completely classify the degrees of r.e. bases of r.e. vector spaces in terms of weak truth table degrees. The ideas extend to classify the degrees of complements and splittings. Several ramifications of the classification are discussed, together with an analysis of the structure of the degrees of pairs of r.e. summands of r.e. spaces.