The Mathematics of Infinity: A Guide to Great Ideas, 2nd Edition

Chapter 7

Inductions and Numbers

Let us consider the well-ordered set from a new perspective. We used two properties of to define well-ordered sets. They were the Trichotomy Property and the Minimum Property. It is accepted by Mathematicians that these properties together with the Principle of Mathematical Induction will give us all a common intellectual picture of the natural numbers. At this stage of intellectual development, the professionals agree that everyone who reads these statements will envision the same set of natural numbers that are right this minute dancing in your head. Furthermore, these principles can be extended to well-ordered sets to give us a new and powerful tool or argument about well-ordered sets. That new tool is called Transfinite Induction. Transfinite Induction is to well-ordered sets what Mathematical Induction is to the natural numbers.

7.1 Mathematical Induction

We begin with a discussion of the Principle of Mathematical Induction or more briefly Mathematical Induction. Intuitively, Mathematical Induction allows us to make a statement for all of the natural numbers by knowing only that the statement holds at 0 and then the statement holds at n + 1 if it holds at n. For instance, suppose you are playing a board game (a thought experiment) whose object ...