Shafer, P orcid.org/0000-0001-5386-9218 (2017) Honest elementary degrees and degrees of relative provability without the cupping property. Annals of Pure and Applied Logic, 168 (5). pp. 1017-1031. ISSN 0168-0072
Abstract
An 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.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2016 Elsevier B.V. This is an author produced version of a paper published in Annals of Pure and Applied Logic. Uploaded in accordance with the publisher's self-archiving policy. |
Keywords: | Sub-recursive hierarchies;; Honest elementary degrees; Degrees of relative provability |
Dates: |
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Institution: | The University of Leeds |
Academic Units: | The University of Leeds > Faculty of Engineering & Physical Sciences (Leeds) > School of Mathematics (Leeds) > Pure Mathematics (Leeds) |
Depositing User: | Symplectic Publications |
Date Deposited: | 07 Feb 2017 11:46 |
Last Modified: | 12 Nov 2017 01:38 |
Published Version: | https://doi.org/10.1016/j.apal.2016.11.005 |
Status: | Published |
Publisher: | Elsevier Masson |
Identification Number: | 10.1016/j.apal.2016.11.005 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:111927 |