## On constructing completions.(English)Zbl 1099.03044

This paper is based on the constructive set theory CZF of P. Aczel [in: Logic colloquium ’77, Proc., Wroclaw 1977, Stud. Logic Found. Math. 96, 55–66 (1978; Zbl 0481.03035)]. If $$A$$ and $$B$$ are sets, a total relation $$r$$ between $$A$$ and $$B$$ is defined to be a subset of $$A\times B$$ such that, for every $$x$$ in $$A$$, there is $$y$$ in $$B$$ with $$(x,y)\in r$$. The class of such relations is denoted $$\text{mv}(A,B)$$. The authors use the following refinement principle, provable in CZF: For sets $$A$$ and $$B$$, there is a set $$C\subseteq \text{mv}(A,B)$$ such that, for every $$r\in \text{mv}(A,B)$$, there is $$s\in C$$ with $$s\subseteq r$$. From this, they deduce that the Dedekind cuts in an ordered set form a set. In particular, the Dedekind reals form a set. A further generalization is obtained by using Richman’s method for completing an arbitrary metric space without sequences to show that the completion of a separable metric space is a set even if the space is a proper class. Thus, every complete separable metric space is a set.

### MSC:

 03E70 Nonclassical and second-order set theories 03F65 Other constructive mathematics 54E35 Metric spaces, metrizability

Zbl 0481.03035
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### References:

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