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 $n$ , $nx \leq y$ . 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 $\mathbb {R}$ 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 $\mathbb {R}$ . 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

^ 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.
^ Counterexamples in Analysis by Bernard R. Gelbaum and John M. H. Olmsted, Chapter 1, Example 7, page 17.

[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.

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