## The Natural Numbers

In this chapter, the authors describe the process of formalizing set theory, so that it supports both the special sets which they shall take as the natural numbers and provide a framework for establishing their familiar properties. They focus on a different definition of natural numbers in terms of sets and logic. The authors provide the basis for almost every proof about the natural numbers. They define operations of addition, multiplication and exponentiation on the set of natural numbers, and show that these operations have the properties that they expect from everyday mathematics. The authors establish properties of multiplication before those of exponentiation, given that the definition of exponentiation exploits multiplication. They prove the pigeon-hole principle for natural numbers. The authors define 'finite', so that 'infinite' will be defined as 'not finite'. Multiplication is the more complicated operation, as it is defined in terms of the more basic operation of addition.