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On constructing completions

Published online by Cambridge University Press:  12 March 2014

Laura Crosilla ,
Hajime Ishihara  and
Peter Schuster
Laura Crosilla
Affiliation:
Università Di Firenze, Dipartimento Di Filosofia, Via Bolognese 52, Firenze 50139, ItalyE-mail:, Laura.Crosilla@unifi.it
Hajime Ishihara
Affiliation:
Japan Advanced Institute for Science and Technology, School of Information Science, Tatsunokuchi, Ishikawa 923–1292, JapanE-mail:, ishihara@jaist.ac.jp
Peter Schuster
Affiliation:
Universität München, Mathematisches Institut, Theresienstr. 39, Munchen 80333, GermanyE-mail:, Peter.Schuster@mathematik.uni-muenchen.de
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Abstract

The Dedekind cuts in an ordered set form a set in the sense of constructive Zermelo–Fraenkel set theory. We deduce this statement from the principle of refinement, which we distill before from the axiom of fullness. Together with exponentiation, refinement is equivalent to fullness. None of the defining properties of an ordering is needed, and only refinement for two–element coverings is used.

In particular, the Dedekind reals form a set: whence we have also refined an earlier result by Aczel and Rathjen, who invoked the full form of fullness. To further generalise this, we look at Richman's method to complete an arbitrary metric space without sequences, which he designed to avoid countable choice. The completion of a separable metric space turns out to be a set even if the original space is a proper class: in particular, every complete separable metric space automatically is a set.

Type
Research Article
Information
The Journal of Symbolic Logic , Volume 70 , Issue 3 , September 2005 , pp. 969 - 978
Copyright
Copyright © Association for Symbolic Logic 2005

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