On the constructive Dedekind reals

Logic and Analysis 1 (2):131-152 (2008)
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Abstract

In order to build the collection of Cauchy reals as a set in constructive set theory, the only power set-like principle needed is exponentiation. In contrast, the proof that the Dedekind reals form a set has seemed to require more than that. The main purpose here is to show that exponentiation alone does not suffice for the latter, by furnishing a Kripke model of constructive set theory, Constructive Zermelo–Fraenkel set theory with subset collection replaced by exponentiation, in which the Cauchy reals form a set while the Dedekind reals constitute a proper class

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Michael Rathjen
University of Leeds

Citations of this work

Logics of intuitionistic Kripke-Platek set theory.Rosalie Iemhoff & Robert Passmann - 2021 - Annals of Pure and Applied Logic 172 (10):103014.
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Topological inductive definitions.Giovanni Curi - 2012 - Annals of Pure and Applied Logic 163 (11):1471-1483.
On the Cauchy completeness of the constructive Cauchy reals.Robert S. Lubarsky - 2007 - Mathematical Logic Quarterly 53 (4‐5):396-414.

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References found in this work

Constructivism in mathematics: an introduction.A. S. Troelstra - 1988 - New York, N.Y.: Sole distributors for the U.S.A. and Canada, Elsevier Science Pub. Co.. Edited by D. van Dalen.
Independence results around constructive ZF.Robert S. Lubarsky - 2005 - Annals of Pure and Applied Logic 132 (2-3):209-225.
On constructing completions.Laura Crosilla, Hajime Ishihara & Peter Schuster - 2005 - Journal of Symbolic Logic 70 (3):969-978.
On the Cauchy completeness of the constructive Cauchy reals.Robert S. Lubarsky - 2007 - Mathematical Logic Quarterly 53 (4‐5):396-414.

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