Dihoum, E and Rathjen, M (2019) Preservation of choice principles under realizability. Logic Journal of the IGPL, 27 (5). pp. 746-765. ISSN 1367-0751
Abstract
Especially nice models of intuitionistic set theories are realizability models V(A)V(A), where AA is an applicative structure or partial combinatory algebra. This paper is concerned with the preservation of various choice principles in V(A)V(A) if assumed in the underlying universe VV, adopting Constructive Zermelo–Fraenkel as background theory for all of these investigations. Examples of choice principles are the axiom schemes of countable choice, dependent choice, relativized dependent choice and the presentation axiom. It is shown that any of these axioms holds in V(A)V(A) for every applicative structure AA if it holds in the background universe.1 It is also shown that a weak form of the countable axiom of choice, ACω,ωACω,ω, is rendered true in any V(A)V(A) regardless of whether it holds in the background universe. The paper extends work by McCarty (1984, Realizability and Recursive Mathematics, PhD Thesis) and Rathjen (2006, Realizability for constructive Zermelo–Fraenkel set theory.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © The Author(s) 2019. Published by Oxford University Press. All rights reserved. This is a pre-copyedited, author-produced version of an article accepted for publication in Logic Journal of the IGPL following peer review. The version of record is available online at: https://doi.org/10.1093/jigpal/jzz002 |
Keywords: | Intuitionistic; constructive Zermelo–Fraenkel set theory; axioms of choice; realizability; applicative structure |
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: | 14 Jan 2019 15:05 |
Last Modified: | 08 Feb 2020 01:38 |
Status: | Published |
Publisher: | Oxford University Press |
Identification Number: | 10.1093/jigpal/jzz002 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:140933 |