6 research outputs found
On the density of honest subrecursive classes
Get PDFThe relation of honest subrecursive classes to the computational complexity of the functions they contain is briefly reviewed. It is shown that the honest subrecursive classes are dense under the partial ordering of set inclusion. In fact, any countable partial ordering can be embedded in the gap between an effective increasing sequence of honest subrecursive classes and an honest subrecursive class which is properly above the sequence (or in the gap between an eflective decreasing sequence and a class which is properly below the sequence). Information is obtained about the possible existence of least upper bounds (greater lower bounds) of increasing (decreasing) sequences of honest subrecursive classes. Finally it is shown that for any two honest subrecursive classes, one properly containing the other, there exists a pair of incomparable honest subrecursive classes such that the greatest lower bound of the pair is the smaller of the first two classes and the least upper bound of the pair is the larger of the first two classes
Honest elementary degrees and degrees of relative provability without the cupping property
Get PDFAn element a of a lattice cups to an element b>ab>a if there is a c<bc<b such that a∪c=ba∪c=b. An element of a lattice has the cupping property if it cups to every element above it. We prove that there are non-zero honest elementary degrees that do not have the cupping property, which answers a question of Kristiansen, Schlage-Puchta, and Weiermann. In fact, we show that if b is a sufficiently large honest elementary degree, then b has the anti-cupping property, which means that there is an a with 0<Ea<Eb0<Ea<Eb that does not cup to b. For comparison, we also modify a result of Cai to show, in several versions of the degrees of relative provability that are closely related to the honest elementary degrees, that in fact all non-zero degrees have the anti-cupping property, not just sufficiently large degrees
