)It is not true that R is the "only" uniformly complete ordered field, but it is the only uniformly complete "Archimedean field", and indeed one often hears the phrase "complete Archimedean field" instead of "complete ordered field".Every uniformly complete Archimedean field must also be Dedekind-complete (and vice versa), justifying using "the" in the phrase "the complete Archimedean field".