Chen, R-M and Rathjen, M (2012) Lifschitz realizability for intuitionistic Zermelo-Fraenkel set theory. Archive for Mathematical Logic, 51 (7-8). 789 - 818 (20). ISSN 0933-5846
Abstract
A variant of realizability for Heyting arithmetic which validates Church's thesis with uniqueness condition, but not the general form of Church's thesis, was introduced by Lifschitz (Proc Am Math Soc 73:101-106, 1979). A Lifschitz counterpart to Kleene's realizability for functions (in Baire space) was developed by van Oosten (J Symb Log 55:805-821, 1990). In that paper he also extended Lifschitz' realizability to second order arithmetic. The objective here is to extend it to full intuitionistic Zermelo-Fraenkel set theory, IZF. The machinery would also work for extensions of IZF with large set axioms. In addition to separating Church's thesis with uniqueness condition from its general form in intuitionistic set theory, we also obtain several interesting corollaries. The interpretation repudiates a weak form of countable choice, AC , asserting that a countable family of inhabited sets of natural numbers has a choice function. AC is validated by ordinary Kleene realizability and is of course provable in ZF. On the other hand, a pivotal consequence of AC , namely that the sets of Cauchy reals and Dedekind reals are isomorphic, remains valid in this interpretation. Another interesting aspect of this realizability is that it validates the lesser limited principle of omniscience.
Metadata
Item Type: | Article |
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Authors/Creators: |
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Copyright, Publisher and Additional Information: | © 2012, Springer. This is an author produced version of a paper published in Archive for Mathematical Logic. Uploaded in accordance with the publisher's self-archiving policy |
Keywords: | Intuitionistic set theory, Lifschitz realizability, Church's thesis, countable axiom of choice, lesser limited principle of omniscience |
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: | 28 Feb 2013 10:51 |
Last Modified: | 03 Nov 2017 22:01 |
Published Version: | http://dx.doi.org/10.1007/s00153-012-0299-2 |
Status: | Published |
Publisher: | Springer |
Identification Number: | 10.1007/s00153-012-0299-2 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:74959 |