44 Chapter 1 • The Real Number System
- Let X be a nonempty set and suppose f, g : X ---+ IR are functions whose
ranges are bounded sets of real numbers. Prove that
(a) sup{f(x) + g(x): x EX}:::; sup{f(x): x EX} +sup{g(x): x EX}.
(b) inf{f(x) + g(x): x EX} 2: inf{f(x): x EX}+ inf{g(x): x EX}. - 7 * .. The" Complete Ordered Field
In this section we take a closer look at Definition 1.6. 13. In Sections 1.1-1.6
we explored successively the properties of fields, ordered fields, Archimedean
ordered fields, and complete ordered fields. We defined a complete ordered field
as a set F together with two operations, + and ·, which satisfy fifteen axioms:
(AO)- (A4), (MO)- (M4), (D), (01)-(03), and (C). We subsequently showed that
a complete ordered field, if there is one, must have the properties we expect
our real number system to have.
Philosophically, two questions of great significance remain to be addressed:
- QUESTION #1: Is there a complete ordered field?
Whenever a concept is defined by specifying the properties one wishes it
to possess, one runs the danger of being overly prescriptive. We may have
listed so many properties that it is impossible for anything to exist that
possesses all these properties. The properties may even be contradictory.
Consider an example from everyday life: Were you to search for a mate
who possessed all the properties that a perfect mate would have, it is
doubtful that you would ever find such a person! Your search might well
end in frustration: sooner or later you could reach the conclusion that you
have been overprescriptive. That is precisely what lies behind Question
#1, in the case of complete ordered fields. We must determine whether we
have laid down so many axioms that nothing exists that satisfies them!
The standard way of dealing with this question is to build up, or "con-
struct,'' a complete ordered field from basic building blocks whose existence is
not questioned, using a process that is beyond reproach. Then, in effect, we
can say that since we accept the existence of the basic building blocks, and
do not question the construction process, we must accept the existence of the
complete ordered field that is constructed by this process.
This constructive process has been carried out to the satisfaction of math-
ematicians, starting with the natural numbers 1, 2, 3, · · · as the basic building
blocks, and using formal logic and set theory as the process. Such a construc-
tion process, however, is deep and more complicated than appropriate for this
course. It would take several weeks of hard and tedious effort. Nevertheless,
an enterprising student may wish to investigate this process as a project. The