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

TL;DR

This paper demonstrates the existence of non-zero honest elementary degrees lacking the cupping property, answering a prior open question and contrasting with related degrees of relative provability.

Contribution

It proves that certain honest elementary degrees do not have the cupping property, including for large degrees, and compares this with degrees of relative provability where the property always holds.

Findings

Existence of non-zero honest elementary degrees without the cupping property.

Large honest elementary degrees can lack the cupping property.

In degrees of relative provability, the property holds for all non-zero degrees.

Abstract

An element $a$ of a lattice cups to an element $b > a$ if there is a $c < b$ such that $a \cup c = b$. 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 $\mathbf b$ is a sufficiently large honest elementary degree, then there is a non-zero honest elementary degree $\mathbf a <_{\mathrm E} \mathbf b$ that does not cup to $\mathbf b$. For comparison, we modify a result of Cai to show that in several versions of the related degrees of relative provability the preceding property holds for all non-zero $\mathbf b$, not just sufficiently large $\mathbf b$.