It will be shown that there exist computably enumerable degrees a and b such that a $>$ b, and for any computably enumerable degree u, if u $\leq$ a and u is cappable, then u $<$ b.

We show the decidability of the existential theory of the recursively enumerable degrees in the language of Turing reducibility, Turing reducibility of the Turing jumps, and least and greatest element.

We employ techniques related to Lempp and Lerman's “iterated trees of strategies” to directly measure a Σ5-predicate and use this in showing the index set of the cuppable r.e. sets to be Σ5-complete. We also show how certain technical devices arise naturally out of the iterated-trees context, in particular, links arise as manifestations of a generalized notion of “stage”.

We employ techniques related to Lempp and Lerman's “iterated trees of strategies” to directly measure a Σ5-predicate and use this in showing the index set of the cuppable r.e. sets to be Σ5-complete. We also show how certain technical devices arise naturally out of the iterated-trees context, in particular, links arise as manifestations of a generalized notion of “stage”.

A completely mitotic computably enumerable degree is a c.e. degree in which every c.e. set is mitotic, or equivalently in which every c.e. set is autoreducible. There are known to be low, low2, and high completely mitotic degrees, though the degrees containing non-mitotic sets are dense in the c.e. degrees. We show that there exists an upper cone of c.e. degrees each of which contains a non-mitotic set, and that the completely mitotic c.e. degrees are nowhere dense in the c.e. (...) degrees. We also show that there is a set computably enumerable in and above 0′ which is not the jump of any completely mitotic degree. (shrink)

We give a proof of a theorem of Harrington that there is no orbit of the lattice of recursively enumerable sets containing elements of each nonzero recursively enumerable degree. We also establish some degree theoretical extensions.

A splitting of an r.e. set A is a pair A1, A2 of disjoint r.e. sets such that A1 A2 = A. Theorems about splittings have played an important role in recursion theory. One of the main reasons for this is that a splitting of A is a decomposition of A in both the lattice, , of recursively enumerable sets and in the uppersemilattice, R, of recursively enumerable degrees . Thus splitting theor ems have been used to obtain results about (...) the structure of , the structure of R, and the relationship between the two structures. Furthermore it is fair to say that questions about splittings have often generated important new technical developments in recursion theory. In this article we survey most of the results and techniques associated with splitting properties of r.e. sets in ordinary recursion theory. (shrink)

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 operators which take a set X to a set relatively computably enumerable in and above X by studying which such sets X can be so mapped into the Turing degree of K. We introduce notions of nontriviality for such operators, and use these to study which additional properties can be required of sets which can be completed to the jump by given operators of this kind.

In this article we establish the existence of a number of new orbits in the automorphism group of the computably enumerable sets. The degree theoretical aspects of these orbits also are examined.

In this article we establish the existence of a number of new orbits in the automorphism group of the computably enumerable sets. The degree theoretical aspects of these orbits also are examined.

Calhoun, W.C., Incomparable prime ideals of recursively enumerable degrees, Annals of Pure and Applied Logic 63 39–56. We show that there is a countably infinite antichain of prime ideals of recursively enumerable degrees. This solves a generalized form of Post's problem.

We consider the set of jumps below a Turing degree, given by JB={x':x≤a}, with a focus on the problem: Which recursively enumerable degrees a are uniquely determined by JB? Initially, this is motivated as a strategy to solve the rigidity problem for the partial order R of r.e. degrees. Namely, we show that if every high2 r.e. degree a is determined by JB, then R cannot have a nontrivial automorphism. We then defeat the strategy—at least in the form presented—by constructing (...) pairs a0, a1 of distinct r.e. degrees such that JB=JB within any possible jump class {x:x'=c}. We give some extensions of the construction and suggest ways to salvage the attack on rigidity. (shrink)