The Algebra of the Natural Numbers

This chapter details the well ordering principle of the natural numbers. It gives some notation and formal mathematical definition. Any collection of objects is a mathematical set. Things in the collection are elements of the set. A complete set of the basic principles that characterize the natural numbers is: there is a unique first natural number; every natural number has a unique immediate successor; every natural number except the first has a unique immediate predecessor; every natural number is an eventual successor of the first; and If S is a set of natural numbers with at least one element, then S has a minimum. Algebraic properties allow us to study and investigate the natural numbers themselves. The algebraic properties are used in the chapter to give direct proofs of many algebraic results. The chapter shows a proof that uses mathematical induction. Mathematical induction is important to clearly understand the logic of an induction proof.