A postdoctoral fellow once laughingly told me, “Every mathematician needs to construct the real numbers at least once.” For most people in science and engineering, the existence of the real numbers is obvious. And that is a good thing; mathematics should describe things as they are or must be, not as we would like to see them. For the mathematician, though, a course in Real Analysis—which one can think of as an exercise in proving the veracity of calculus from the ground up—is often a crash course in not taking anything for granted. Sure, the real numbers are an important mathematical framework in engineering and data science, but is this framework the unique, simplest representation?
In this blog post, we are going to construct the real numbers from nothing more than set theory.
Furthermore, we are going to do so in a way that shows how the real numbers are unique in the properties we would want from such a number system. We will make a very simple characterization of sets, and from this characterization we will construct first the natural numbers (i.e., “0, 1, 2, …”), then the integers, then the rational numbers, and lastly the reals.
Ready, Set, Go!
For our purposes, a set is an object which can “contain” or “point to” any object other than itself (if you're interested in why we require a set to not contain itself, read up on Russell's Paradox!). By saying that it “can” contain objects, we mean to imply the existence of a completely empty set, which we denote with the “∅” symbol. If a set S consists only of the objects a, b, and c, we can write S = {a, b, c}. Therefore, because there exists the empty set ∅, there also exists a set S = {∅} containing the empty set and nothing else. Similarly, there must exist a set S’ = {S, ∅}, a set S’’ = {S’, S, ∅}, a set S* = {S’’, S}, and so on and so forth (this might remind you of a certain Christopher Nolan film). We end up constructing infinitely many sets in this way, and so there necessarily exist sets with infinitely many elements as well.
So, now we have sets. Hooray! Where are all the numbers?
Natural numbers are…natural.
How do the natural numbers arise from the sets we just constructed? How can we even talk about “finite” sets without having natural numbers to begin with? The answer might surprise you; we can call a set S infinite if there exists a map from S to a proper subset of itself, and we call it finite otherwise. We are not going to prove this here, but a way to motivate this is to think about how the positive integers Z+ contain the positive, even integers Z2+ as a proper subset, even though multiplication by 2 maps the elements of Z+ to Z2+. This definition of finiteness completely agrees with the conventional notion of finite numbers. Further, the natural numbers arise as the equivalence classes of finite sets under set bijection.
Integers: making a semigroup whole
Our construction of the natural numbers as the equivalence classes of finite sets induces an interesting property. If sets A and B belong to equivalence classes m and n, respectively, and they share no common elements, then the union of A and B belongs to the equivalence class (m+n). This “addition” we just defined comprises a semigroup operation: a rule for composing two objects and an identity element preserving objects (For any set S, the union of S and ∅ is S, so the equivalence class containing ∅ acts like adding zero to a number).
You probably read the word “semigroup” and thought, “What does a whole group look like?” A group is simply a semigroup such that, given a and b, there always exists a unique x in the semigroup satisfying the equation a + x = b. When b happens to be the identity element, we call x the inverse of a with respect to the semigroup operation. In the case of sets, this notion of inverse composition gives us a notion of removing elements from sets while preserving the semigroup property. In this way, we have found that the integers are the smallest group containing the natural number semigroup under the law of addition.
Rationals: if you liked it, then you should have put a field on it
Like how the union of disjoint sets induces a semigroup structure on the natural numbers, the Cartesian product of two sets induces a distinct semigroup structure on the natural numbers—and by extension, the integers. Given sets S and S’, the Cartesian product S x S’ of S and S’ is the set of all pairs of objects (a, b) for a in S and b in S’. If the respective cardinalities of S and S’ are m and n, then we can say that S x S’ belongs to the equivalence class m x n in a way that agrees completely with the conventional notion of integer multiplication.
(The Cartesian product might seem conveniently contrived here, but it is an important product in topology and tensor algebra. To motivate this: think of how three-dimensional space is simply the Cartesian product of three copies of the real line.)
So now, we can endow the integers with not just an additive group structure, but a multiplicative semigroup structure as well! This comprises what mathematicians call a ring. What happens when we add inverses to the multiplicative group? Then the ring becomes what we call a field: one that happens to be the smallest field containing the ring.
(We can create such an inverse for all elements in the ring besides the additive identity. Play around and see what happens otherwise!)
Reals: the limit of perfection
The rational numbers are pretty fantastic; being a mathematical field, there’s a sense in which they are already complete. Why would we need anything else?
Many of you are familiar with the Fibonacci sequence: the sequence of numbers (f0, f1, f2, …) defined by initial condition (f0, f1) = (0, 1) and the recurrence fn = fn-1 + fn-2. Fibonacci and others before him observed that the sequence of ratios rn = fn / fn-1 stabilizes, but does not converge to any rational number. François E. A. Lucas discovered a separate sequence—with initial condition (L0, L1) = (1, 3) and the same recurrence relation—which produces a sequence of ratios 𝞺n = Ln / Ln-1 with an interesting property: the sequence (r0 - 𝞺0, r1 - 𝞺1, …) also converges, and in particular converges to zero. So, in the set of convergent sequences of rational numbers, there exist equivalence classes of sequences which do not converge to any rational number, though there are of course sequences converging to q for all rational q. So, in order to fill all “holes” we observe when we look at convergent sequences, it is necessary to define equivalence classes of convergent sequences, whose addition and multiplication can be defined by pairwise operations. The equivalence class that results—together with the field structure inherited from the addition and multiplication—is the real numbers.
Back out of the rabbit hole
This exercise was a high-level way to demonstrate how the work of rigorous mathematics is to formalize the consequences of a small set of axioms. On a more personal level, I think it demonstrates how much beauty and precision permeate everything that mathematics touches.
Thank you for taking this short journey with me! I would encourage you to take a course in Real Analysis if this sort of thing fascinates you. If I were grading you for your efforts here, I would certainly give you an A (Unless, of course, you did not understand the Christopher Nolan reference. In that case, you get a B.).
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