- 7 *"The" Complete Ordered Field 45
classic reference^10 is [82]; other constructions may be found in Chapter 1 of
Part B of [10], as well as in [11], [25], [37], [44], [63], [74], [96], [117], and [129].
An easier way to deal with Question #1 is to merely assume the following
as an axiom:AXIOM: There exists a complete ordered field.In effect, this is the approach we take in this book. We simply assume
this axiom, realizing that, as we have just discussed, it would be possible to
construct a complete ordered field, starting only with a set of axioms for the
natural number system. This decision seems appropriate for this course.
- QUESTION #2: Is there more than one complete ordered field?
Students of mathematics become very familiar with defining a concept by
specifying its properties. For example, we are familiar with the process of
defining groups, vector spaces, rings, and so on. Anything that satisfies
certain properties is a "group,'' for example. We ordinarily expect there
to be many different examples that satisfy these properties. In fact, be-
cause there are many different groups, we feel it is necessary to categorize
them into classes of look-alikes. We use the concept of isomorphism to
express this alikeness. Two algebraic systems are isomorphic as groups,
for example, if they are both groups and are indistinguishable in the sense
that one of the groups is merely a relabeling of the other. The concern
raised by Question #2 is this: When we categorize complete ordered fields
into classes according to isomorphism, do we get more than one class? We
should be quite surprised if we get only one isomorphism class. This has
never happened for other algebraic systems that we have encountered.
(Think of groups, rings, fields, vector spaces, etc.)
It can be shown by algebraic methods that any two complete ordered
fields are isomorphic! Of course that statement requires clarification. We
shall not prove this fact in this course, but will state it as a formal theorem
and provide a guideline for proving it as a project.
Theorem 1.7.1 (Uniqueness of the Complete Ordered Field) If F and
F' are any two complete ordered fields, then 3 a 1-1 correspondence f : F ---> IQ;.fi!HI
F' 3
(1) "ix, y E F, J(x + y) = J(x) + J(y);
(2) "ix, y E F , J(xy) = J(x) · J(y);
(3) f (OF) =OF';
- The numbers in square brackets refer to entries in the Bibliography, which follows Ap-
pendix B.