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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

Citations:

Zbl 0481.03035
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References:

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