Order in the Reals

Abstract

You’ve seen numbers ever since you’ve been in school, and you know a lot about them. It is possible to give them a careful mathematical foundation. In fact, it’s possible to construct the natural numbers (and you can do so in Project 29.3). Then, if you try to introduce operations like addition and subtraction, you’ll find that you are missing something: the negative numbers. So you look at the integers, and try again. Now, trying to introduce multiplication and division, you’ll find you are missing something again: multiplicative inverses. So you look at the rational numbers, and you’ll find you are missing something yet again. That brings you to the real numbers. Our ultimate goal will be to discuss what’s missing in ℚ, and to show you why ℝ has what’s missing. This is known as completeness of ℝ. In this and the next chapter, we’ll show you some wonderful applications of completeness.