Non-Archimedean ordered field

In mathematics, a non-Archimedean ordered field is an ordered field that does not satisfy the Archimedean property. Such fields will contain infinitesimal and infinitely large elements, suitably defined.

Definition

Suppose F is an ordered field. We say that F satisfies the Archimedean property if, for every two positive elements x and y of F, there exists a natural number n such that nx > y. Here, n denotes the field element resulting from forming the sum of n copies of the field element 1, so that nx is the sum of n copies of x.

An ordered field that does not satisfy the Archimedean property is a non-Archimedean ordered field.

Examples

The fields of rational numbers and real numbers, with their usual orderings, satisfy the Archimedean property.

Examples of non-Archimedean ordered fields are the Levi-Civita field, the hyperreal numbers, the surreal numbers, the Dehn field, and the field of rational functions with real coefficients (where we define f > g to mean that f(t)>g(t) for large enough t).

Infinite and infinitesimal elements

In a non-Archimedean ordered field, we can find two positive elements x and y such that, for every natural number n, nx ≤ y. This means that the positive element y/x is greater than every natural number n (so it is an "infinite element"), and the positive element x/y is smaller than 1/n for every natural number n (so it is an "infinitesimal element").

Conversely, if an ordered field contains an infinite or an infinitesimal element in this sense, then it is a non-Archimedean ordered field.

Applications

Hyperreal fields, non-Archimedean ordered fields containing the real numbers as a subfield, are used to provide a mathematical foundation for nonstandard analysis.

Max Dehn used the Dehn field, an example of a non-Archimedean ordered field, to construct non-Euclidean geometries in which the parallel postulate fails to be true but nevertheless triangles have angles summing to π.^[1]

The field of rational functions over ${\displaystyle \mathbb {R} }$ can be used to construct an ordered field that is Cauchy complete (in the sense of convergence of Cauchy sequences) but is not the real numbers.^[2] This completion can be described as the field of formal Laurent series over ${\displaystyle \mathbb {R} }$. It is a non-Archimedean ordered field. Sometimes the term "complete" is used to mean that the least upper bound property holds, i.e. for Dedekind-completeness. There are no Dedekind-complete non-Archimedean ordered fields. The subtle distinction between these two uses of the word complete is occasionally a source of confusion.

References

1. Dehn, Max (1900), "Die Legendre'schen Sätze über die Winkelsumme im Dreieck", Mathematische Annalen, 53 (3): 404–439, doi:10.1007/BF01448980, ISSN 0025-5831, JFM 31.0471.01, S2CID 122651688.
2. Counterexamples in Analysis by Bernard R. Gelbaum and John M. H. Olmsted, Chapter 1, Example 7, page 17.