Rathjen, M (2012) Constructive zermelo-fraenkel set theory, power set, and the calculus of constructions. In: Dybjer, P, Lindström, S, Palmgren, E and Sundholm, G, (eds.) Epistemology versus ontology: Essays on the philosophy and foundations of mathematics in honour of Per Martin-Löf. Logic, Epistemology, and the Unity of Science, 27 . Springer , Dordrecht, Netherlands , 313 - 349. ISBN 9789400744356
Abstract
Full intuitionistic Zermelo-Fraenkel set theory, IZF, is obtained from constructive Zermelo-Fraenkel set theory, CZF, by adding the full separation axiom scheme and the power set axiom. The strength of CZF plus full separation is the same as that of second order arithmetic, using a straightforward realizability interpretation in classical second order arithmetic and the fact that second order Heyting arithmetic is already embedded in CZF plus full separation. This paper is concerned with the strength of CZF augmented by the power set axiom, CZFP. It will be shown that it is of the same strength as Power Kripke-Platek set theory, KP(P), as well as a certain system of type theory, MLVP, which is a calculus of constructions with one universe. The reduction of CZFP to KP(P) uses a realizability interpretation wherein a realizer for an existential statement provides a set of witnesses for the existential quantifier rather than a single witness. The reduction of KP(P) to CZFP employs techniques from ordinal analysis which, when combined with a special double negation interpretation that respects extensionality, also show that KP(P) can be reduced to CZF with the negative power set axiom. As CZF augmented by the latter axiom can be interpreted in MLVP and this type theory has a types-as-classes interpretation in CZFP, the circle will be completed.
Metadata
Item Type: | Book Section |
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Authors/Creators: |
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Editors: |
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Copyright, Publisher and Additional Information: | © 2012, Springer. This is an author produced version of a book chapter published by Springer. Uploaded in accordance with the publisher's self-archiving policy. The final publication is available at www.springerlink.com |
Keywords: | Constructive set theory; Martin-Lof type theory; Mac Lane set theory; realizability with sets of witnesses; Power Kripke-Platek set theory; calculus of constructions; set recursion; power recursion; negative power set axiom; proof-theoretic strength |
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 09:40 |
Last Modified: | 28 Oct 2016 00:36 |
Published Version: | http://dx.doi.org/10.1007/978-94-007-4435-6 |
Status: | Published |
Publisher: | Springer |
Series Name: | Logic, Epistemology, and the Unity of Science |
Identification Number: | 10.1007/978-94-007-4435-6 |
Open Archives Initiative ID (OAI ID): | oai:eprints.whiterose.ac.uk:75182 |