“Math is about nothing,” Andrew Hacker says in a recent New Yorker article by Rebecca Mead. If he had not already, with this statement he has certainly jumped the shark. He follows it up even more perplexingly: “Math describes much of the world but is all about itself,” he says. So math is about nothing, except what it is about, which is math and some things in the world? I echo math teacher Patrick Honner, who asks "Why are we listening to Andrew Hacker?"

Hacker’s assertion is absurd, but today let’s humor him. Meet the empty set.






OK, the empty set is not exactly the most photogenic of sets. It’s a set that has nothing inside, and it’s kind of hard to get a good picture of nothing. The two most popular depictions of the set are empty brackets {} and something that looks like the Scandinavian vowel ø. (The Wikipedia page for the letter warns us not to confuse the letter with the mathematical symbol, but it does not specify what dire consequences we will face if we stumble.) I will use the symbol ∅ because I think it looks nicer, and when you’re writing about nothing, it might as well be well-dressed nothing.

In a recent post on his blog Mathematics Without Apologies, Michael Harris says that ∅ is hardly anyone’s favorite set, but I think there are a few reasons to give it some love.


The first reason to love the empty set is that, far from limiting your possibilities, the empty set opens them up through the magic of vacuous truth. Anything is true if you start reasoning from a false premise. We’re secretly using this idea rhetorically when we say “I’ll enter a digits of pi reciting contest when pigs fly” or “If that’s a sensible rail ticket pricing scheme, then I’m the Queen of England.” 

The empty set is the prototypical generator of vacuous truths. Anything you want to prove about things in the empty set, you can. Logically, this is equivalent to saying “if x is in the empty set, then x has [whatever delightful property you’re thinking about].” Do you want a unicorn? Great! Everything in the empty set is a unicorn. Do you hate unicorns? You’re in luck—everything in the empty set is a talisman against unicorns. The empty set itself is not a talisman against unicorns, but every last thing in it is.

The empty set is ubiquitous in mathematics, and I mean that literally. It is a subset of every other set. Here, we have to be careful about what we mean by a subset. A set X is a subset of a set Y if every element of X is an element of Y. And when X is the empty set, this is true no matter what Y is! Due to the power of vacuous truth, every element of the empty set is an even number, and every element of the empty set is an odd number. So the empty set is a subset of both the even numbers and the odd numbers.


I think the fact that the empty set is a subset of every set makes mathematical emptiness fundamentally different from the way we think about emptiness in everyday life. When you pour water into an empty glass, it feels like you’ve removed the emptiness from the glass. But you can’t take the emptiness out of mathematical sets. 

I had a fun conversation with Laura Taalman, a mathematician at James Madison University, about the empty set. She zeroed (seewhatIdidthere?) in on the difference between plain old emptiness and the empty set. How could we experience the empty set in our lives? An empty meal would not just be a time during the day when you weren’t eating. You would sit down in front of a place setting, no one would bring you food, and then you’d get up and leave. 4'33", John Cage’s famous empty composition, is not just any 273 seconds of quiet. The performer sits at a piano in front of an audience. The wrapper is important.

That wrapper helps explain a curious but crucial fact about mathematical emptiness: the empty set is different from the set containing the empty set. Sometimes people compare it to plastic bags or empty glasses. The empty set is one empty glass, while the set containing the empty set is a stack of two glasses. You can keep going: put the glasses into a bucket, and you’ve got a set (the bucket) containing the set containing the empty set (the glasses). Put another glass in there next to them, and you’ve got a set containing (the empty set and the set containing the empty set). Here language starts to get a bit ambiguous, so I'm supplementing with extra punctuation. Of course, analogies to physical objects aren’t perfect. A bucket with three glasses in is a thing with some heft to it, while mathematical sets do not come in weighty containers.

Something from nothing

The most glamorous thing about the empty set is probably the way you can use it to create something from nothing. Before we do that, we might want to take a brief step back and think about set theory. Most of modern mathematics is founded on set theory. A set is basically the most general collection of things a mathematician can talk about. If you have a bunch of things and call them a group, that means the things have to satisfy some rules about their relationships with each other. A set, on the other hand, can be any collection of things, no relationships required. The set {green things, a capybara named Millicent, 6, honesty} is just as valid a set as the whole numbers. Of course, when we get to actually doing math, we quickly run out of interesting things to do with my rather eclectic collection, while the whole numbers have provided generations of mathematicians with problems to ponder.

Within set theory, mathematicians have settled on Zermelo-Fraenkel set theory as the rock upon which mathematics is built. Most of us never work at that foundational level, but in principle we could tunnel down from our latest theorem all the way to the ZF axioms. There are a few different formulations of those axioms that are equivalent in the sense that you can get the axioms of one formulation from the axioms of the other. In all of them, the empty set is necessary. In fact, it’s the only set we’re guaranteed to have. When your only tool is a hammer, every problem looks like a nail, and the empty set is our hammer.

To build the whole numbers from the empty set, we start small. We let the empty set be 0, or ∅ =0. Then the set containing the empty set is 1. We can write that as either {0} or {∅}. Then 2 is {0,1}, or {∅,{∅}}; 3 is {0,1,2}, or {∅,{∅},{∅,{∅}}}. Notice that at each stage, the number of elements is the same as the number we’re defining. That’s a little hard to tell with 3, but notice that the last comma is within a single element of the set. As a set, the number 3 has 3 things in it: the empty set; the set containing the empty set; and the set containing the empty set and the set containing the empty set.

By the time we get to 4, things are pretty intense. 4={0,1,2,3} or {∅;{∅};{∅,{∅}};{∅,{∅},{∅,{∅}}}}. In words, it’s like buffalo buffalo buffalo: it’s the set containing: the empty set; the set containing the empty set; the set containing the empty set and the set containing the empty set; and the set containing the empty set, the set containing the empty set, and the set containing the set containing the empty set and the set containing the empty set. Here’s one place where using notation rather than language makes something completely baffling into something you can almost hold onto. This construction isn’t the only way to make numbers in set theory, but it’s probably the most famous one.


It looks like we’ve built whole numbers, from which we can derive the rest of modern mathematics, from the empty set. Is Hacker right? Is mathematics about nothing? Here’s the truth: mathematics didn’t evolve out of set theory. Most mathematicians never even think about building numbers out of sets, or most of the nuts and bolts of set theory in general. Set theory is barely over a hundred years old, but people have been doing mathematics for millennia. Math did not develop axiomatically. It developed piecemeal—though often with striking similarities—all over the world because people had problems they needed to solve or noticed patterns they wanted to describe and analyze.

We are inquisitive and experimental creatures, and the meaning of mathematics gradually shifted to the more rigidly axiomatized system we use today, but we are still using it to describe the world and solve problems. Some of the ideas that seem the most esoteric—packing spheres in eight dimensions, let’s say—have ties to incredibly practical questions—finding the best error-correcting codes for data in cellular networks or fiber-optic cables so you, I, or Andrew Hacker can communicate with each other even in a world full of distortion and noise.

If that’s about nothing, the bar to something is awfully high.

Read about more of my favorite spaces:
The Cantor Set
Fat Cantor Sets
The Topologist’s Sine Curve
Cantor's Leaky Tent
The Infinite Earring
The Line with Two Origins
The House with Two Rooms
The Fano Plane
The Torus
The Three-Torus
The Möbius Strip
The Long Line
Space-Filling Curves
The Wallis Sieve
Two Tori Glued along a Slit