Certify Ordered Field For Free

Theorem. There exists a complete ordered field R, called the real numbers. In fact, R is the unique1 complete ordered field.

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 Number System is a set of numbers that includes all rational and irrational numbers. It is a complete ordered field, meaning that it satisfies certain properties like closure under addition and multiplication, total ordering, and the existence of supremum and infimum for bounded sets.

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.

Definition of an ordered field: An ordered field is a field containing a subset of elements closed under addition and multiplication and having the property that every element in the field is either 0, in the subset, or has its additive inverse in the subset.

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.

Any set which satisfies all eight axioms is called a complete ordered field. We assume the existence of a complete ordered field, called the real numbers. The real numbers are denoted by R.

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.