Advanced Mathematics

4.2The Complete Ordered Field: The Real Numbers

Purpose of Section

We present the axiomatic definition of the real numbers as a complete ordered field. We carry out this three‐step process by first defining an algebraic field, then introduce an order on the algebraic field, yielding an ordered field, and then the completeness axiom that allows us to define the complete ordered field, of which there is only one, the real numbers.

4.2.1 Introduction

Advances in function theory in the nineteenth century demanded a deeper understanding of the real numbers, which led to a “rigorization” of analysis by such mathematical greats as Cauchy, Abel, Dedekind, Dirichlet, Weierstrass, Bolzano, Frege, Cantor, and others. A deeper understanding of functions required precise proofs which in turn required the real number system be placed on solid mathematical ground.

Although we generally think of real numbers as points on a continuous line that extends endlessly in both directions, the goal of this chapter is to strip away everything you know about the real numbers and start afresh. This is not easy since all knowledge and mental imagery of the real numbers created over a lifetime is firmly entrenched in our minds. But if the reader is willing to wipe the slate clean and start anew, we will introduce you to a new mathematical entity, known by mathematicians as the complete, ordered field, which, for lack of another name, we call ℝ. By building the axioms of the real numbers, you will ...