# Archimedean property

## Primary tabs

Synonym:
axiom of Archimedes, Archimedean principle
Type of Math Object:
Theorem
Major Section:
Reference

## Mathematics Subject Classification

### Axiom vs. Theorem

While the phrase "is neither an axiomn nor attributed to Archimedes"
is fine rhetoric, the first half of it needs to be taken with a
teaspoon of salt. Whether or not a statement is an axiom depends
on how one chooses to axiomatize one's theory. To be sure, if one
chooses the least upper bound property as an axiom, then the
Archimedean property follows as a consequence, hence is a theorem.

However, there is more than one way to axiomatize the theory of
real numbers. For instance, I could also characterize the real
number system as follows:

* The real number system is an Archimedean ordered field.

* No proper extension field of the real number system is Archimedean.

In this case, the roles are reversed --- the Archimedean property is
now an axiom and the least upper bound property is now a theorem.