The Dedekind cuts in an ordered set form a set in the sense of constructiveZermelo—Fraenkel set theory. We deduce this statement from the principle ofrefinement, which we distill before from the axiom of fullness. Togetherwith exponentiation, refinement is equivalent to fullness. None of thedefining properties of an ordering is needed, and only refinement fortwo—element coverings is used.
In particular, the Dedekind reals form a set; whence we have also refined anearlier result by Aczel and Rathjen, who invoked the full form of fullness.To further generalise this, we look at Richman's method to complete anarbitrary metric space without sequences, which he designed to avoidcountable choice. The completion of a separable metric space turns out to bea set even if the original space is a proper class; in particular, everycomplete separable metric space automatically is a set.
"On constructing completions." J. Symbolic Logic 70 (3) 969 - 978, September 2005. https://doi.org/10.2178/jsl/1122038923