Journal of Symbolic Logic

On constructing completions

Laura Crosilla, Hajime Ishihara, and Peter Schuster

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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.

Article information

Source
J. Symbolic Logic, Volume 70, Issue 3 (2005), 969-978.

Dates
First available in Project Euclid: 22 July 2005

Permanent link to this document
https://projecteuclid.org/euclid.jsl/1122038923

Digital Object Identifier
doi:10.2178/jsl/1122038923

Mathematical Reviews number (MathSciNet)
MR2155275

Zentralblatt MATH identifier
1099.03044

Citation

Crosilla, Laura; Ishihara, Hajime; Schuster, Peter. On constructing completions. J. Symbolic Logic 70 (2005), no. 3, 969--978. doi:10.2178/jsl/1122038923. https://projecteuclid.org/euclid.jsl/1122038923


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