72 + 42: Characterizations of the Completeness and Archimedean Properties of Ordered Fields

Abstract

This paper provides a list of statements of single-variable Real Analysis, including well-known theorems, that are equivalent to the completeness or Archimedean properties of totally ordered fields. There are 72 characterizations of completeness and 42 characterizations of the Archimedean property, among them many that appear to be new in the sense that they do not seem to have previously been mentioned in this context. An attempt is made to be as comprehensive as possible and to give a complete account of the current state of knowledge of the matter. Proofs are provided whenever they are not readily available in the literature.