Archimedean Property - Examples and Non-examples - Equivalent Definitions of Archimedean Ordered Field

Equivalent Definitions of Archimedean Ordered Field

Every linearly ordered field K contains (an isomorphic copy of) the rationals as an ordered subfield, namely the subfield generated by the multiplicative unit 1 of K, which in turn contains the integers as an ordered subgroup, which contains the natural numbers as an ordered monoid. The embedding of the rationals then gives a way of speaking about the rationals, integers, and natural numbers in K. The following are equivalent characterizations of Archimedean fields in terms of these substructures.

1. The natural numbers are cofinal in K. That is, every element of K is less than some natural number. (This is not the case when there exist infinite elements.) Thus an Archimedean field is one whose natural numbers grow without bound.

2. Zero is the infimum in K of the set {1/2, 1/3, 1/4, … }. (If K contained a positive infinitesimal it would be a lower bound for the set whence zero would not be the greatest lower bound.)

3. The set of elements of K between the positive and negative rationals is closed. This is because the set consists of all the infinitesimals, which is just the closed set {0} when there are no nonzero infinitesimals, and otherwise is open, there being neither a least nor greatest nonzero infinitesimal. In the latter case, (i) every infinitesimal is less than every positive rational, (ii) there is neither a greatest infinitesimal nor a least positive rational, and (iii) there is nothing else in between, a situation that points up both the incompleteness and disconnectedness of any non-Archimedean field.

4. For any x in K the set of integers greater than x has a least element. (If x were a negative infinite quantity every integer would be greater than it.)

5. Every nonempty open interval of K contains a rational. (If x is a positive infinitesimal, the open interval (x, 2x) contains infinitely many infinitesimals but not a single rational.)

6. The rationals are dense in K with respect to both sup and inf. (That is, every element of K is the sup of some set of rationals, and the inf of some other set of rationals.) Thus an Archimedean field is any dense ordered extension of the rationals, in the sense of any ordered field that densely embeds its rational elements.