ordered field

In fact, it is a very special ordered field: the biggest one.

Let S be a vector space over the real numbers, or, more generally, some ordered field.

In mathematics, there are several ways of defining the real number system as an ordered field.

The real numbers form an ordered field which is Archimedean and order complete.

Every ordered field can be embedded into the surreal numbers.

Its order and algebraic structure make it into an ordered field.

For example, it is not enough to construct an ordered field with infinitesimals.

The real numbers are often described as "the complete ordered field", a phrase that can be interpreted in several ways.

Every ordered field is a formally real field (see below).

More generally, the definition applies to a vector space over an ordered field.

There are two equivalent definitions of an ordered field.

The synthetic approach gives a list of axioms for the real numbers as a complete ordered field.

The complex numbers do not form an ordered ring (or ordered field).

Any Dedekind-complete ordered field is isomorphic to the real numbers.

This field cannot be turned into an ordered field.

Nevertheless, archaeology did become a more ordered field, in contrast to the individual and sporadic efforts that came before.

In particular, if every M is an ordered field, then so is the ultraproduct.

An ordered field has some additional nice properties.

Hence the theory of the real ordered field with restricted analytic functions is model complete.

Ordered fields that have infinitesimal elements are also called non-Archimedean.

Dubois showed in 1967 that the answer is negative in general for ordered fields.

This provides a connection between surreal numbers and more conventional mathematical approaches to ordered field theory.

Most importantly, the real numbers form an ordered field, in which addition and multiplication preserve positivity.

An ordered field is a field F together with a positive cone P.

In abstract algebra, it can be shown that any complete ordered field is isomorphic to the real numbers.