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.

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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.

v t e Infinitesimals
History	Adequality Leibniz's notation Integral symbol Criticism of nonstandard analysis The Analyst The Method of Mechanical Theorems Cavalieri's principle
Related branches	Nonstandard analysis Nonstandard calculus Internal set theory Synthetic differential geometry Smooth infinitesimal analysis Constructive nonstandard analysis Infinitesimal strain theory (physics)
Formalizations	Differentials Hyperreal numbers Dual numbers Surreal numbers
Individual concepts	Standard part function Transfer principle Hyperinteger Increment theorem Monad Internal set Levi-Civita field Hyperfinite set Law of continuity Overspill Microcontinuity Transcendental law of homogeneity
Mathematicians	Gottfried Wilhelm Leibniz Abraham Robinson Pierre de Fermat Augustin-Louis Cauchy Leonhard Euler
Textbooks	Analyse des Infiniment Petits Elementary Calculus Cours d'Analyse

This page was last edited on 2 March 2024, at 05:05

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