Suppose that we define ℝ abstractly instead of by explicit construction, i.e. we just say that ℝ is any Dedekind-complete* ordered field. Can we now prove that ℝ is a complete metric space? Does the question even make sense? I mean, the definition of "metric space" refers to ℝ. What ℝ is that anyway, the abstract one or one defined by explicit construction (Dedekind cuts)? *) By that I mean that every set that's bounded from above has a least upper bound.