An Archimedean field (or ordered ring, were someone to use such a beast) is an ordered field with the following property: (Underlined letters are elements of the field, which, being ordered, contains all natural numbers.)

Any complete ordered field which is Archimedean is (isomorphic to) the real numbers.

Archimedes was the first mathematician to NOTICE this axiom; he did so in the context of geometry, but the name stuck to the general case.