What makes a number system a number system? In this book the real numbers serve as the standard with which other number systems are compared. To be called a number system a mathematical system must share most if not all of the fundamental properties of the reals.

What are the fundamental properties of the reals? We use a set of properties (or laws or axioms) that characterize the reals completely, meaning that any mathematical system with these properties is the same as the reals. Such a set of properties for a particular mathematical object is called a categorical axiom system. Many such systems are known. A famous one for plane geometry goes back to Euclid, although a correct and complete categorical axiom system for Euclidean geometry was formulated only late in the 19th century. In this later period and on into the 20th century there has been tremendous interest in axiom systems and their application to all areas of mathematics.

Part One of this book describes a categorical axiom system for the reals. Chapter One lists the axioms of this system. Chapter Two constructs the reals (from the rational numbers), and shows that they satisfy the axioms presented in Chapter One. In addition, we prove that any mathematical system satisfying these axioms is identical (more technically, isomorphic) to the reals.

A categorical axiom system is a powerful tool. The one we describe is used in this book to analyze and compare number systems. Given any system we ask: which axioms for the reals does it satisfy? The answer reveals how close the new system is to the standard set by the reals themselves.