Definition talk:Dedekind Completeness Property

Is this a different concept from Definition:Complete Metric Space or are they equivalent? If not, is there a connection? There are subtleties here I haven't adequately explored. --prime mover (talk) 07:50, 22 September 2012 (UTC)

I would think that this is a different concept. Dedekind completeness applies to ordered sets; Cauchy completeness applies to metric spaces. The axiomatic definition of the real numbers $\R$ that I know of is that it is the unique Dedekind complete totally ordered field, up to isomorphism (should be added someday). Given the axioms for $\R$ as a totally ordered field, Dedekind completeness is equivalent to Cauchy completeness and the Archimedean property together. But if Dedekind completeness is replaced by Cauchy completeness alone, then there are several non-isomorphic structures that satisfy the modified axioms for $\R$. One of them is the field $\map \Q {\left({\epsilon}\right)}$ of formal Laurent series over the field $\Q$ of rational numbers (with the total ordering on $\Q \left({\left({\epsilon}\right)}\right)$ defined appropriately), which is neither Dedekind complete nor Archimedean. --abcxyz (talk) 18:20, 22 September 2012 (UTC)