Dedekind cuts of Archimedean complete ordered abelian groups

A Dedekind cut of an ordered abelian group G is a pair (X, Y) of nonempty subsets of G where Y=G−X and every member of X precedes every member of Y. A Dedekind cut (X, Y) is said to be continuous if X has a greatest member or Y has a least member, but not both; if every Dedekind cut of G is a continuous cut, G is said to be (Dedekind) continuous. The ordered abelian group R of real numbers is, of course, up to isomorphism the unique (Dedekind) continuous ordered abelian group. R is also up to isomorphism the unique Archimedean complete, Archimedean ordered abelian group. The idea of an Archimedean complete ordered abelian group was introduced by Hans Hahn [17] as a generalization of Hilbert’s [19, 20] classical continuity condition which characterizes R as an Archimedean ordered field which admits no proper extension to an Archimedean ordered field.