In mathematics, a non-Archimedean ordered field is an

ordered field In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. Basic examples of ordered fields are the rational numbers and the real numbers, both with their standard ord ...

that does not satisfy the

Archimedean property In abstract algebra and mathematical analysis, analysis, the Archimedean property, named after the ancient Greek mathematician Archimedes of Syracuse, Italy, Syracuse, is a property held by some algebraic structures, such as ordered or normed g ...

. Such fields will contain infinitesimal and infinitely large elements, suitably defined.

Definition

Suppose is an

. We say that satisfies the

if, for every two positive elements and of , there exists a

natural number In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive in ...

such that . Here, denotes the field element resulting from forming the sum of copies of the field element , so that is the sum of copies of . An ordered field that does not satisfy the Archimedean property is a non-Archimedean ordered field.

Examples

The fields of

rational number In mathematics, a rational number is a number that can be expressed as the quotient or fraction of two integers, a numerator and a non-zero denominator . For example, is a rational number, as is every integer (for example, The set of all ...

s and

real number In mathematics, a real number is a number that can be used to measure a continuous one- dimensional quantity such as a duration or temperature. Here, ''continuous'' means that pairs of values can have arbitrarily small differences. Every re ...

s, with their usual orderings, satisfy the Archimedean property. Examples of non-Archimedean ordered fields are the

Levi-Civita field In mathematics, the Levi-Civita field, named after Tullio Levi-Civita, is a non-Archimedean ordered field; i.e., a system of numbers containing infinite and infinitesimal quantities. It is usually denoted \mathcal. Each member a can be constructed ...

, the

hyperreal number In mathematics, hyperreal numbers are an extension of the real numbers to include certain classes of infinite and infinitesimal numbers. A hyperreal number x is said to be finite if, and only if, , x, for some integer

n

s, the

surreal number In mathematics, the surreal number system is a total order, totally ordered proper class containing not only the real numbers but also Infinity, infinite and infinitesimal, infinitesimal numbers, respectively larger or smaller in absolute value th ...

s, the Dehn field, and the field of

rational function In mathematics, a rational function is any function that can be defined by a rational fraction, which is an algebraic fraction such that both the numerator and the denominator are polynomials. The coefficients of the polynomials need not be ...

s with real coefficients (where we define to mean that for large enough ''t'').

Infinite and infinitesimal elements

In a non-Archimedean ordered field, we can find two positive elements and such that, for every natural number , . This means that the positive element is greater than every natural number (so it is an "infinite element"), and the positive element is smaller than for every natural number (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 The history of calculus is fraught with philosophical debates about the meaning and logical validity of fluxions or infinitesimal numbers. The standard way to resolve these debates is to define the operations of calculus using (ε, δ)-definitio ...

Max Dehn Max Wilhelm Dehn (November 13, 1878 – June 27, 1952) was a German mathematician most famous for his work in geometry, topology and geometric group theory. Dehn's early life and career took place in Germany. However, he was forced to retire in 1 ...

used the Dehn field, an example of a non-Archimedean ordered field, to construct

non-Euclidean geometries In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those that specify Euclidean geometry. As Euclidean geometry lies at the intersection of metric geometry and affine geometry, non-Euclidean geo ...

in which the

parallel postulate In geometry, the parallel postulate is the fifth postulate in Euclid's ''Elements'' and a distinctive axiom in Euclidean geometry. It states that, in two-dimensional geometry: If a line segment intersects two straight lines forming two interior ...

fails to be true but nevertheless triangles have angles summing to . The field of rational functions over

\R

can be used to construct an ordered field that is

Cauchy complete In mathematical analysis, a metric space is called complete (or a Cauchy space) if every Cauchy sequence of points in has a limit that is also in . Intuitively, a space is complete if there are no "points missing" from it (inside or at the bou ...

(in the sense of convergence of Cauchy sequences) but is not the real numbers.''Counterexamples in Analysis'' by Bernard R. Gelbaum and John M. H. Olmsted, Chapter 1, Example 7, page 17. This completion can be described as the field of

formal Laurent series In mathematics, a formal series is an infinite sum that is considered independently from any notion of convergence, and can be manipulated with the usual algebraic operations on series (addition, subtraction, multiplication, division, partial su ...

over

\R

. It is a non-Archimedean ordered field. Sometimes the term "complete" is used to mean that the

least upper bound property In mathematics, the least-upper-bound property (sometimes called completeness, supremum property or l.u.b. property) is a fundamental property of the real numbers. More generally, a partially ordered set has the least-upper-bound property if ever ...

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

{{Infinitesimal navbox Ordered algebraic structures Real algebraic geometry Nonstandard analysis