ABSTRACT
In this chapter, the authors begin by considering some simple mathematics problems that students in elementary or middle school might be asked to solve. In addition to the arithmetic axioms, there are also axioms that describe the way the integers are ordered, and how this ordering interacts with addition and multiplication. If the Division Algorithm seems obvious, it's probably because its proof relies on an equally obvious axiom called the Well-Ordering Principle, which states that every nonempty subset of the natural numbers has a smallest element. The Division Algorithm is also useful when studying the idea of congruence, which is used by mathematicians to describe cyclic phenomena in the world of the integers. In fact, the most common modern construction of the integers is based entirely on sets and set operations.
