Non-Archimedean ordered field

In mathematics, a non-Archimedean ordered field is an ordered field that does not satisfy the Archimedean property. Examples are the Levi-Civita field, the hyperreal numbers, the surreal numbers, the Dehn field, and the field of rational functions with real coefficients with a suitable order.

Definition

The Archimedean property is a property of certain ordered fields such as the rational numbers or the real numbers, stating that every two elements are within an integer multiple of each other. If a field contains two positive elements x < y for which this is not true, then x/y must be an infinitesimal, greater than zero but smaller than any integer unit fraction. Therefore, the negation of the Archimedean property is equivalent to the existence of infinitesimals.

Applications

Hyperreal fields, non-Archimedean ordered fields containing the real numbers as a subfield, may be used to provide a mathematical foundation for non-standard 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/2.^[1]

The field of rational functions over $\R$ can be used to construct an ordered field which is complete (in the sense of convergence of Cauchy sequences) but is not the real numbers.^[2] Sometimes the term complete is used to mean that the least upper bound property holds. With this meaning of complete there are no complete non-Archemedian 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, http://books.google.com/books?id=vEbWAAAAMAAJ&pg=PA404 .
2. ^ Counterexamples in Analysis by Bernard R. Gelbaum and Bernard R. Gelbaum, Chapter 1, Example 7, page 15.