Honest elementary degrees without the cupping property

Paul Shafer, University of Leeds. Part of the logic seminar series.

An element a of a lattice cups to an element b > a if there is a c < b such that a join c is b. An element of a lattice has the cupping property if it cups to every element above it. We study cupping in the lattice of honest elementary degrees, in which functions with elementary recursive graphs are compared via the elementary-recursive-in relation. Kristiansen showed that every sufficiently large honest elementary degree has the cupping property. This result prompted Kristiansen, Schlage-Puchta, and Weiermann to ask if in fact every non-zero honest elementary degree has the cupping property. We answer their question negatively by showing that if b is a sufficiently large honest elementary degree, then there is a non-zero honest elementary degree a < b that does not cup to b. Building on work of Cai, we also prove non-cupping results for the degrees of relative provability. Finally, we discuss the connections among the honest elementary degrees, the degrees of relative provability, and provability in Peano arithmetic and its fragments.

Paul Shafer, University of Leeds