Math 3040 Spring 2009 The Natural Numbers

Proofs are used both to assist or insure the solution of specific problems in mathematics and to provide the cement that holds together the construction of large-scale mathematical structures, such as the theory of real numbers or the theory of groups. In this and subsequent chapters, we shall be using proofs for a large-scale project, namely the construction of some of the most common number systems used in mathematics. We shall begin by constructing the natural numbers and then follow a classical path through the construction of the integers and rationals. The path then branches, with one branch leading to the development of the real numbers and the beginnings of analysis and the other with an algebraic focus. In part, this project may serve as an introduction to some important mathematical ideas. But in equal measure, it is intended as a display of proof techniques and of how proofs are used to build mathematical theories.