Archimedean axiom

An axiom, originally formulated for segments, which states that if the smaller one of two given segments is marked off a sufficient number of times, it will always produce a segment larger than the larger one of the original two segments. This axiom can be formulated in an analogous manner for surfaces, volumes, positive numbers, etc. In general, the Archimedean axiom applies to a given quantity if for any two values $A$ and $B$ of this quantity such that $A<B$ it is always possible to find an integer $m$ such that $Am>B$. The axiom forms the base of the process of successive division in arithmetic and in geometry (cf. Euclidean algorithm). The importance of the Archimedean axiom only became fully apparent after the discovery, in the 19th century, of magnitudes to which it does not apply — so-called non-Archimedean quantities (cf. Quantity; see also Archimedean group; Archimedean ring; Archimedean class).

The axiom was explicitly formulated by Archimedes (3rd century B.C.) in his work The sphere and the cylinder; it had been previously utilized by Eudoxus of Cnides, and is for this reason also called Eudoxus' axiom.