ABSTRACT
In this chapter, we examine Peano’s fifth axiom, also known as the Axiom of Induction, and explore the consequences of this axiom. Among the consequences are four versions of the Principle of Mathematical Induction, the Well-Ordering Principle for the set of the natural numbers, and the Fundamental Theorem of Arithmetic. It is difficult, if not impossible, to determine when the very first proof by induction was employed. In 1202, Leonardo Pisano Bigollo (1180-1250), also known as Leonardo of Pisa and after his death called Fibonacci, used induction in his Book of the Abacus to prove 6(12 + 22 + 32 + · · ·+ n2) = n(n+ 1)(n+ n+ 1). In 1321, Levi Ben Gershon (1288-1344) completed The Art of the Calculator in which several propositions were proven using mathematical induction. In 1654, the French mathematician Blaise Pascal (1623-1662) gave the first definitive explanation of the method of mathematical induction. However, the name “mathematical induction” was not associated with this method of proof until August De Morgan (1806-1871) published his article on “Induction (Mathematics)” in the Penny Cyclopaedia of 1838.
