Excerpt from The Real Number System

Why are these properties true? Where do they come from? What other properties do the real numbers have? Just what is a real number anyway?

The present volume tells about real numbers by investigating the kind of structure that 13 formed by the totality of all real numbers. The real number system is described categorically in the, book by a set of axioms, stated brie?y: the real number system is a complete ordered field. These axioms are presented and studied step by step, with numerous examples used to illustrate the new concepts as they are introduced. Examples of ordered fields that are not complete, fields that are not ordered, and systems that are similar to fields but are not fields are all examined with a View to an increased under standing of the real number system itself. The discussion is extensive and multiple-leveled, and is designed for more than one particular audience. As a monograph the book attempts to tell a fairly-full story, including much about induction and going all the way to the laws of exponents for arbitrary positive bases and real exponents. An appendix shows how the property of completeness can be described in a large number of equivalent fashions.

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