Accept Ordered Field For Free

Definition. A complete ordered field is an ordered field F with the least upper bound property (in other words, with the property that if S ? F, S = ? and S is bounded above then S has a least upper bound supS). Example 14. The real numbers are a complete ordered field.

We say that a ring R is ordered if there exists a non-empty subset P ? R satisfying the following: ?a, b ? P, a + b ? P and a · b ? P. (Trichotomy) Exactly one of the following holds ?a ? R: a ? P, a = 0,?a ? P. This set P is meant to model the positive integers in Z, and is called the subset of positive elements of R.

In mathematics, an ordered field is a field together with a total ordering of its elements that is compatible with the field operations. The basic example of an ordered field is the field of real numbers, and every Dedekind-complete ordered field is isomorphic to the reals.

Rational numbers are an ordered field The integers do not form a field! They almost do though, but just don't have multiplicative inverses (except that the integer 1 is its own multiplicative inverse ? it is also the multiplicative identity).

An ordered set is a set S in which an order is defined. He also defined order in his book: Let S be a set. An order on S is a relation, denoted by <, with the following two properties: If x, y ? S then one and only one of x < y, x = y, x > y is true.

A set F together with two operations + and ? and a relation < satisfying the 13 axioms above is called an ordered field. Thus the real numbers are an example of an ordered field. Another example of an ordered field is the set of rational numbers Q with the familiar operations and order.

The real numbers are an ordered field, with P = {x ? R : x > 0}.