Construction of the
Number Svstems of
CHAPTER 10
In Chapter 9 we studied the number systems N, 2, Q, R, and C from a
descriptive point of view. For most applications requiring knowledge of
elementary properties of the real numbers and associated number systems,
the material in that chapter provides adequate information. There is, how-
ever, a totally different approach to the real numbers, a constructive ap-
proach, by which we literally build the real number system from more basic
number systems. An in-depth study of this approach is a formidable and
lengthy project, and is not necessarily appropriate for all sophomore and
junior mathematics students. An understanding of the general structure
and of a number of crucial details of this approach, though, probably @
worthwhile and well within the capability of those who have successfully
progressed through the preceding chapters of this text. We propose to
present such a general outline in this final chapter.
For reasons philosophical and historical, as well as mathematical, every
serious student of abstract mathematics should be at least generally familiar
with an approach to constructing the reals. Whether one uses the approach
of Dedekind cuts or (as we do) Cauchy sequences to pass from the rationals
to the reals, the general approach is the same. The reals are constructed
from the rationals, which, in turn, have been constructed from the integers,
with the latter having been built from the positive integers. Tlie starting
point for all these constructions is an axiomatic description of the natural
number system such as the axiomatization of Peano.