Honest elementary degrees and degrees of relative provability without the cupping property

Honest elementary degrees and degrees of relative provability without the cupping property

An element a of a lattice cups to an element b > a if there is a c < b such that a∪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 [17]. In fact, we show that if b is a sufficiently ...

On the structure of honest elementary degrees

We present some new results, and survey some old results, on the structure of honest elementary degrees. This paper should be a suitable first introduction to the honest elementary degrees.

On the Structure of the Degrees of Relative Provability

We investigate the structure of the degrees of provability, which measure the proof-theoretic strength of statements asserting the totality of given computable functions. The degrees of provability can also be seen as an extension of the investigation of relative consistency statements for firstorder arithmetic (which can be viewed as Π1-statements, whereas statements of totality of computable ...

Universal Cupping Degrees

Cupping nonzero computably enumerable (c.e. for short) degrees to 0′ in various structures has been one of the most important topics in the development of classical computability theory. An incomplete c.e. degree a is cuppable if there is an incomplete c.e. degree b such that a∪b = 0′, and noncuppable if there is no such degree b. Sacks splitting theorem shows the existence of cuppable degrees....

Cupping Classes of Enumeration Degrees