Approve Ordered Field For Free

An ordered field F is complete if every nonempty subset T of F that is bounded above has a least upper bound supT ? F. Theorem. There exists a complete ordered field R, called the real numbers. In fact, R is the unique1 complete ordered field.

Familiar examples of ordered sets include the number systems ?, ?, ? and ? with their usual orders, as well as spaces of real valued functions, ordered by the pointwise order.

Answer and Explanation: The integers and natural numbers are ordered, but are not fields since they do not contain multiplicative inverses (the natural numbers also don't include the additive identity 0, or additive inverses). The complex numbers are a field, but cannot be ordered.

The set of real numbers has a field structure, under the operations of ordinary addition and ordinary multiplication. The set of real numbers is also a totally ordered set. Taken together, these facts are almost enough to mean the real numbers form an ordered field.

We prove that any two complete ordered fields are isomorphic to one another. Put differently: R is the only complete ordered field (up to isomorphism). In a typical introductory real analysis course, one defines R to be a complete ordered field.

There are lots of examples of fields, some of which are ordered fields. However, we will see that there is, up to isomorphism, only one complete ordered field. By ?up to isomorphism?, we mean that if R and S are both complete ordered fields, then there is an isomorphism from R to S.

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).