We all work with real numbers every day, although different people use different subsets. Everybody uses the integers, numbers with no fractional part such as 2, -158, or 0, to count things out or to figure out how many minutes are left until a certain time. We also use rational numbers (which are fractions of integers such as -5/7 or 11/3 as well as the integers, since 11 = 11 / 1) when calculating prices: a price of $5.99 is the same as 599/100 dollars, and finding a 15% tip is the same as multiplying by 15/100. Mathematicians use real numbers, which are the rationals as well as the square root of 2 or ?, the ratio of the circumference of a circle to its diameter. But most people (including many mathematicians!) don’t really know what the reals are, only that they have certain properties. That’s what this series, ‘Constructing the Reals’, is about: constructing a solid definition of the real numbers starting with only the concept of sets, and in the process proving wrong the famous mathematician Leopold Kronecker, who once remarked that “God made the integers; all else is the work of man.” As we will see, even the integers are ‘the work of man’; God only made the empty set!
Of course, you can’t go straight from sets to real numbers; you have to make stops along the way. There are different orders in which various subsets of the reals can be constructed, but one common way is to start with the non-negative integers (usually, but not always!, called the natural numbers), then construct the non-negative rational numbers, then all rational numbers, and finally all real numbers. This is the path that I’ll take, with one construction per post.

A word of caution is required here: as we construct larger subsets of reals, the ‘same’ number will appear as different sets. For example, 0 is both the set {}, which contains nothing, and the set that contains all pairs of natural numbers where the first element is 0 and the second is not, such as (0, 2) or (0, 9). To avoid ambiguity, I’ll use subscripts on numbers to distinguish which is being considered: natural numbers will use ?, non-negative rationals will use ?+, general rationals will use ?, and the general real number will not use anything. The same convention applies for operations such as addition and comparisons like less than; <_? refers to comparing natural numbers, as opposed to <_?+ which compares non-negative rationals. It turns out equality is always the same, with two things being equal if and only if they have the same elements, so we don’t need a subscript for those. I’ll also leave out subscripts when there’s no danger of ambiguity. With that out of the way, let’s begin!

The concept of a set is hard to define exactly, but for our purposes we can define it as ‘a collection of other objects without repetition’. The simplest set is the set that contains nothing at all, the empty set, written as {} or ?. So we identify it with the simplest number, 0. The next simplest set is the set that contains the empty set, { {} }, which we call 1. But 2 is not the set that contains 1_?, but rather the set containing both 0 and 1: 2 = {0, 1} = { {}, { {} } }. We continue to add more and more numbers like this, each one being the same as the set of all the numbers before it.

We now have the natural numbers, but we don’t yet know how to do anything with them. Equality is obvious: two numbers are equal if and only if they have the same elements in them. The next obvious operation is ‘less than’: we say x < y if and only if x is in y (or, in set-theory notation,