Frittaion, E. orcid.org/0000-0003-4965-9271, Nemoto, T. orcid.org/0000-0003-3898-6189 and Rathjen, M. orcid.org/0000-0003-1699-4778 (Cover date: October–November 2023) Choice and independence of premise rules in intuitionistic set theory. Annals of Pure and Applied Logic, 174 (9). 103314. ISSN 0168-0072
Abstract
Choice and independence of premise principles play an important role in characterizing Kreisel's modified realizability and Gödel's Dialectica interpretation. In this paper we show that a great many intuitionistic set theories are closed under the corresponding rules for finite types over . It is also shown that the existence property (or existential definability property) holds for statements of the form , where the variable y ranges over objects of finite type σ. This applies in particular to (Constructive Zermelo-Fraenkel set theory) and (Intuitionistic Zermelo-Fraenkel set theory), two systems known not to have the general existence property. On the technical side, the paper uses a method that amalgamates generic realizability for set theory with truth, whereby the underlying partial combinatory algebra is required to contain all objects of finite type.
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Copyright, Publisher and Additional Information: | © 2023 Elsevier B.V. This is an author produced version of an article published in Annals of Pure and Applied Logic. Uploaded in accordance with the publisher's self-archiving policy. | ||||
Keywords: | Intuitionistic set theory; Independence of premise; Axiom of choice; Finite types; Realizability interpretation; Partial combinatory algebras | ||||
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) | ||||
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Depositing User: | Symplectic Publications | ||||
Date Deposited: | 27 Sep 2023 09:07 | ||||
Last Modified: | 27 Sep 2023 09:07 | ||||
Status: | Published | ||||
Publisher: | Elsevier | ||||
Identification Number: | https://doi.org/10.1016/j.apal.2023.103314 | ||||
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