Real numbers and other completions

A notion of completeness and completion suitable for use in the absence of countable choice is developed. This encompasses the construction of the real numbers as well as the completion of an arbitrary metric space. The real numbers are characterized as a complete Archimedean Heyting field, a terminal object in the category of Archimedean Heyting fields. (© 2008 WILEY‐VCH Verlag GmbH & Co. KGaA, Weinheim)