Toward a More Complete List of Completeness Axioms

Abstract

We first discuss the Cut Axiom, due to Dedekind, which is one of the many equivalent formulations of the completeness of the real numbers. We point out that the Cut Axiom is equivalent to four “cornerstone theorems” of single-variable Real Analysis, namely, the Intermediate, Extreme, and Mean Value Theorems, as well as Darboux's Theorem.We then describe some general properties of ordered fields, in particular the Archimedean Property and its consequences, and provide a list of statements that are equivalent to completeness and may thus serve as alternate completeness axioms.