ABSTRACT
Real Numbers This chapter looks at the real number system from an axiomatic point of view, and thereby lays the foundation for our future work. It may seem, at first sight, that we are being too pedantic in spending too much time discussing or proving intuitively obvious ideas. However, we should keep in mind that intuition, though often useful, can sometimes be misleading. A prime example of this is the intuitive idea that any measurable length can be expressed as a common fraction. This led to the assumption that
√ 2 may be represented by a rational number,
which was later proved to be false. This will be discussed in section 2.3. Any attempt to build the real numbers by a constructive approach,
first by defining the natural numbers, then the integers, then the rational numbers, and finally the real numbers, will not be an easy task. See, for example, [RUD] for an exposition of how the real numbers may be constructed from the rationals using Dedekind cuts. We shall, instead, adopt an approach as old as Euclid, whereby we assume the existence of a set R, called the set of real numbers, which satisfies certain properties, called axioms or postulates, that will allow us to derive the other pertinent properties we seek in this course. This approach, it seems to us, is less ambiguous and more economical, and should not obstruct or diminish our intuitive capability if the axioms are chosen carefully enough. There are twelve axioms to be imposed on R, the first eleven of
which are algebraic. These are the field axioms (A1 to A9) and the order axioms (A10 and A11). But the twelfth, the completeness axiom, is different, and it is this last axiom which provides R with a topological closure property that allows us to conduct analysis. The axioms will be presented in three stages in order to emphasize their separate implications.
