330 CONSTRUCTION OF NUMBER SYSTEMS Chapter 10
Philosophically, knowledge of this construction provides insight into the
famous remark of Dedekind to the effect that the positive integers are a
creation of God while all else is the work of man. It also provides back-
ground information to support the statement, in Article 9.4, that the com-
plex numbers are no less "real" than the real numbers.
Historically, this construction emanates from a time, the late nineteenth
century, when the modern idea of the real numbers as a concrete object
(i.e., the unique complete ordered field) had not developed fully. The posi-
tive integers were generally regarded as the "base in reality" for mathema-
tical analysis, and the prevalent view was that a number system such as
the reals had to be built from them in an explicit way in order to be credited
with "existence." Modern mathematics, of course, is perfectly happy to as-
sume existence of the reals as an axiom (our approach in Chapter 9), as
long as it is understood precisely how the system of real numbers is defined.
The construction of the reals, however, is not to be regarded by students
purely as an historical curiosity. It involves a wealth of mathematical ideas
that are important and useful for the study of mathematics today, many
of which are probably new to you. Two main ideas are: (1) the role of the
axiomatic approach in abstract mathematics (Article 1) and (2) the impor-
tance of equivalence classes (recall Chapter 7) in mathematical constructions
(Articles 2 and 3), and thus the value of equivalence classes as a tool in
existence proofs. Students taking courses in formal logic or non-Euclidean
geometry will receive extensive exposure to (I), while students learning
about quotient structures in an abstract algebra course will become familiar
with several key examples of (2). Additional important topics arising in this
chapter are recursive definition, the well-ordering principle, well-definedness
of an algebraic operation, and Cauchy sequence, among others.
As indicated earlier, many of the proofs are omitted from this chapter.
Students who are particularly interested in the ideas presented here may
wish to "work through" the material, filling in missing proofs. Those who
elect only to "read through" the chapter should find the experience profitable
as well, due both to expanding their general knowledge about relationships
among number systems and to exposure to the topics just mentioned.
10.1 An Axiomatization for the
System of Positive IntegersIn this article we take our first steps toward replacing the ad hoc axiom
of Chapter 9, "a complete ordered field exists," with basic assumptions that
are more intuitive and, in a sense, believable. We start with a two-part
axiom, based on the celebrated five postulates of Guiseppe Peano, published
in 1889. At first glance, this axiom appears to have bearing on the system
of positive integers only, but as we will see in Articles 10.2 and 10.3, its
assumption leads to the construction of the systems of integers and rationals