Beyond Infinity

From Uncountable Numbers to a Chicken-Sandwich Sandwich, an Exploration of Math's Biggest Topic


By Eugenia Cheng

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Understanding the concept of infinity is a lofty task, but this creative and easy-to-follow book helps break down all the mathematic complexities so anyone can gain a better understanding of the universe.

"[Cheng] does a great service by showing us non-mathematician schlubs how real mathematical creativity works." — Wall Street Journal

How big is the universe? How many numbers are there? And is infinity + 1 is the same as 1 + infinity? Such questions occur to young children and our greatest minds. And they are all the same question: What is infinity? In Beyond Infinity, Eugenia Cheng takes us on a staggering journey from elemental math to its loftiest abstractions. Along the way, she considers how to use a chessboard to plan a worldwide dinner party, how to make a chicken-sandwich sandwich, and how to create infinite cookies from a finite ball of dough. Beyond Infinity shows how one little symbol holds the biggest idea of all.


I hate airports.

I find airports stressful, crowded, noisy. There are usually too many people, too many queues, not enough seats, and unhealthy food everywhere tempting me to eat it. It's a shame when this is the way traveling starts, as it makes me dread the journey. Traveling should be an exciting process of discovery. Airports – along with cramped economy seating – too often mar what is the almost miraculous and magical process of flying somewhere in a plane.

Mathematics should also be an exciting process of discovery, an almost miraculous and magical journey. But it is too often marred by the way it starts, with too many facts or formulae being thrown at you, and stressful tests and unpalatable problems to solve.

By contrast, I love boat trips.

I love being out on the open water, feeling the wind in my face, watching civilization and the coastline recede into the distance. I like heading toward the horizon without it ever getting any closer. I like feeling some of the power of nature without being entirely at its mercy: I'm not a sailor, so usually someone else is in charge of the boat. Occasionally there are boats I can manage, and then the exertion is part of the reward: a little rowing boat that I once rowed around a small moat encircling a tiny chateau in France; a pedal boat along the canals of Amsterdam; punting on the river Cam, although after once falling in I was put off for life, just like some people are put off mathematics for life by bad early experiences. I have taken boat trips to see magnificent whales off the coast of Sydney and Los Angeles, or seals and other wildlife off the coast of Wales. Then there are the ferries crossing the Channel to France that started most of our family holidays when I was little, before the improbable Eurostar was built. How quickly we humans can come to take something for granted even though it previously seemed impossible!

These days I rarely take boats with the purpose of getting anywhere – rather, the purpose is to have fun, see some sights or some nature, and possibly exert myself. The one exception is the Thames River ferry, which is a very satisfying way to commute into central London, joyfully combining the fun of a boat trip and a journey with a destination.

I love abstract mathematics in somewhat the same way that I love boat trips. It's not just about getting to a destination for me. It's about the fun, the mental exertion, communing with mathematical nature and seeing the mathematical sights. This book is a journey into the mysterious and fantastic world of infinity and beyond. The sights we'll see are mind-boggling, breathtaking, and sometimes unbelievable. We will revel in the power of mathematics without being at its mercy, and we will head toward the horizon of human thinking without that horizon ever getting any closer.

part one


Infinity is a Loch Ness monster, capturing the imagination with its awe-inspiring size but elusive nature. Infinity is a dream, a vast fantasy world of endless time and space. Infinity is a dark forest with unexpected creatures, tangled thickets, and sudden rays of light breaking through. Infinity is a loop that springs open to reveal an endless spiral.

Our lives are finite, our brains are finite, our world is finite, but still we get glimpses of infinity around us. I grew up in a house with a fireplace and chimney in the middle, with all the rooms connected in a circle around it. This meant that my sister and I could chase each other round and round in circles forever, and it felt as if we had an infinite house. Loops make infinitely long journeys possible in a finite space, and they are used for racetracks and particle colliders, not just children chasing each other.

Later my mother taught me how to program on a Spectrum computer. I still smile involuntarily when I think about my favorite program:


20 GOTO 10

This makes an endless loop – an abstract one rather than a physical one. I would hit RUN and feel delirious excitement at watching HELLO scroll down the screen, knowing it would keep going forever unless I stopped it. I was the kind of child who was not easily bored, so I could do this every day without ever feeling the urge to write more useful programs. Unfortunately this meant my programming skills never really developed; infinite patience has strange rewards.

The abstract loop of my tiny but vast program is made by the program going back on itself, and self-reference gives us other glimpses of infinity. Fractals are shapes built from copies of themselves, so if you zoom in on them they keep looking the same. For this to work, the detail has to keep going on "forever," whatever that means – certainly beyond what we can draw and beyond what our eye can see. Here are the first few stages of some fractal trees and the famous Sierpinski triangle.

If you point two mirrors at each other, you see not just your reflection, but the reflection of your reflection, and so on for as long as the angle of the mirrors permits. The reflections inside the reflections get smaller and smaller as they go on, and in theory they could go on "forever" like the fractals.

We get glimpses of infinity from loops and self-reference, but also from things getting smaller and smaller like the reflections in the mirror. Children might try to make their piece of cake last forever by only ever eating half of what's left. Or perhaps you're sharing cake, and everyone is too polite to have the last bite so they just keep taking half of whatever's left. I'm told that this has a name in Japanese: enryo no katamari, the last piece of food that everyone is too polite to eat.

We don't know if the universe is infinite, but I like staring up at a church spire and tricking myself into thinking that the sides are parallel and it's actually an infinite tower soaring up into the sky to infinity. Our lives are finite, but fictional and mythological tales of immortality appear through the ages and across cultures.

So much for glimpses of infinity, like ripples in the waters of Loch Ness that may or may not have been caused by a giant, ancient, mysterious monster. What is this monster we call infinity? What do we mean when we throw around the innocuous-sounding word "forever"? In our modern impatient world, people are prone to using it rather hyperbolically, exclaiming "I've been on hold forever!" after two minutes, or "This web page is taking forever to load!" if it takes more than about three seconds. I'm told by Basque writer Amaia Gabantxo that in Basque the word for eleven is hamaika, but it also means infinity. This is borne out by another friend of mine who did an audit of homemade jam in the cupboard and declared there were "four jars from 2013, ten from 2014, and lots from 2015." Apparently more than ten might as well be infinity. My research field is higher-dimensional category theory, and there "higher" usually means three dimensions or more, including infinity – everything from just three to infinity turns out to be about the same.

The way we think about infinity in our normal lives might be dreamy and exciting, but it dissipates under close examination, like the end of a rainbow, which will never be there if you try to go looking for it. It causes paradoxes and contradictions, impassable ravines and murky traps. It doesn't stand up to the tests of rigorous logic, as we'll see.

One of the roles of mathematics is to explain phenomena in the world around us, especially phenomena that crop up in many different places. If a similar idea relates to many different situations, mathematics swoops in and tries to find an overarching theory that unifies those situations and enables us to better understand the things they have in common. Infinity is one of those ideas. It pops up all over the place as an idea that we can dream about, and seems similar to other ideas that can be unified by mathematics: the ideas of length, size, quantity. So why is it so difficult to extend those easy mathematical concepts to include infinity? This is what this book is about: why it is difficult, how it eventually can be done, and what we see along the way.

Infinity Instincts

Infinity is easy to think about but hard to pin down. Small children can quickly latch on to the idea of infinity, but it took mathematicians thousands of years to work out how to explain infinity in full-technicolor logical glory. Here are some things we might think about infinity. Children come up with these thoughts about infinity quite often, all by themselves:

Infinity goes on forever.

Infinity is bigger than the biggest number.

Infinity is bigger than anything we can think of.

If you add one to infinity it's still infinity.

If you add infinity to infinity it's still infinity.

If you multiply infinity by infinity it's still infinity.

Children can get very excited when the notion of infinity first dawns on them. They learn to count up to ten, and then twenty, and then they learn about a hundred, a thousand, a million, a billion. If you ask a small child what the biggest possible number is, they may well say "a billion," but then you can ask them about a billion and one and watch their eyes widen.

It's not very hard to convince them that no matter what number they think of, you could always add one and get a bigger number. This gives the idea that there is no biggest number. Numbers go on forever! But then, how many numbers are there in total? The idea of infinity starts to emerge.

Perhaps some children first hear of infinity because they watch the Toy Story films and they hear Buzz Lightyear saying, "To infinity… and beyond!," which does sound very exciting. When I was a child Toy Story hadn't been made yet, but I had inklings of infinity from the loops I described earlier, the physical loops in our house and the abstract loops in my favorite computer program.

Once children start thinking about infinity, they can easily come up with questions about it that are teasingly difficult to answer. What is infinity? Is it a number? Is it a place? If it's not a place, how can we go there and beyond?

If children hear about infinity at school, the questions just start proliferating. Is one divided by zero infinity? Is one divided by infinity zero? If infinity plus one is infinity, what happens when you subtract infinity?

When children ask innocuous math questions that seem impossible to answer, this can be intimidating for adults, who feel that they are supposed to have all the answers. But as math educator and innovator Christopher Danielson says, an important aspect of learning is being able to ask new questions; this is more important than being able to state new facts. In math, there are always more questions. Even people who did quite well at math, even people who did math at university, even research mathematicians have more questions about infinity than can be answered.

Infinity Weirdness

Here are some of my favorite mind-boggling conundrums about infinity that we are going to explore.

If you have an infinite hotel and it's full, you can still fit another guest in by moving everyone up one room.

If there were a lottery with infinitely many balls, what would be your chances of winning?

Some infinities are bigger than others!

Infinite pairs of socks are somehow more infinite than infinite pairs of shoes.

If I were immortal, I could procrastinate forever.

If you are traveling from A to B, you first have to cover half the distance, then half the remaining distance, and then half the remaining distance, and so on. There will always be half the remaining distance left, so you'll never get there. Or will you?

The recurring decimal is exactly equal to 1.

Does a circle have infinitely many sides?

Why do people who do quite well at math often get stuck at calculus? Yes, this is a question about infinity too.

Infinity can appeal to people of all ages, of all levels of expertise, in different ways. This book is going to be a journey to infinity – and beyond. Because there really is something beyond infinity, if you think about it hard enough and in the right way. Just like there are always more questions to ask and more things to understand. Infinity is not a physical place, so this is not a physical journey. You can go with me on this journey sitting right where you are, because the journey is an abstract journey. It's a journey into the deep, tangled, mysterious, murky, endless world of ideas.


Why do we go on this journey? Like with physical journeys, there are many reasons to go on an abstract journey. Everyone has their own reasons for traveling. Maybe there's something particular you want to do at your destination. Maybe there's a really good view from the top. Maybe there's beautiful scenery along the way. Maybe you enjoy the physical exertion of walking or climbing, or the exhilaration of riding a bicycle or driving a fast car, or the serenity of sitting on a train watching the countryside rush past. (My experience of trains involves more delays and irate commuters than serenity, but let's ignore that for a second.) Maybe you like going into the unknown. Maybe you like wandering around and losing yourself completely in a city. Maybe you have the travel bug and want to see as much of our incredible earth as possible, simply because there is so much to see.

All of these reasons have their counterparts in the abstract world. A journey where you have something particular to do (like commuting to work) corresponds to having a particular problem you want to solve or a particular application in mind. This type of abstract journey is less about the sense of discovery and more about getting something done. Going up high for a good view is like the abstract investigations we do to gain new perspectives on things we've already seen up close. The beautiful scenery along the way is the weird and wonderful ideas and scenarios that come up as we investigate. And yes, there is exhilaration in the mental exertion, and excitement in thinking about ideas that seem incomprehensible and then slowly clear up, like fog clearing to reveal the ocean glistening to the horizon. I don't have the travel bug for the physical world as much as some people do, but I do have the curiosity bug, the counterpart for the abstract world. I'm fairly calm and resigned about there being a lot of the world I haven't seen, but I am insatiable when it comes to ideas I don't understand. I always want to explore them. As soon as I catch a glimpse of something I don't understand, I am driven to plunge into it. I like losing myself completely in a city, and I like losing myself in ideas as well. As much as I am driven to understand things, I am also happy to acknowledge the things we humans cannot understand. In fact, I positively revel in it. It means that there is always more out there, which is a glorious thing. Wouldn't it be a bit sad if you could finally say yes, that's it, I've been to every restaurant in London now? But of course that's impossible. There will always be another restaurant you haven't tried, and there will always be things we don't understand.

In a strange way this book isn't about infinity at all. It's about the excitement of a journey into the abstract unknown. Jules Verne's Journey to the Center of the Earth wasn't really about the center of the earth, but about the excitement of an incredible journey. This is a book about how abstract thinking works and what it does for us. It's about how it helps us pin down what we really mean when we start having an interesting idea. It doesn't necessarily explain the whole idea: mathematics doesn't explain everything about infinity. But it does help us become clear about what we can and can't do with infinity.

This first part of the book, then, is a journey toward understanding what infinity is. If you ask a small child what they think infinity is, they might say something like "It's bigger than any number you can think of." This is true, but it still doesn't tell us what infinity is, any more than saying "Yao Ming is taller than anyone you've ever met" tells you who Yao Ming is.

In the second part of the book we'll take a tour of the world equipped with our new ideas about infinity and see where this elusive creature has been lurking all along. It's in mirrors that we point at each other, racetracks that we run around, every journey we take, and in every changing situation in this continuously changing world of ours. Understanding infinity is the basis of the field of calculus, which is inextricably embedded in almost every aspect of modern life.

Of course, it is possible to enjoy all those aspects of modern life without having the slightest understanding of calculus, which is why I don't emphasize these applications as the main reason for writing about infinity. Mathematics suffers a strange burden of being required to be useful. This is not a burden placed on poetry or music or football. If you ask me what all this is useful for, I could answer by saying it helps us generate electricity; make phone calls; build bridges, roads, and airplanes; provide water to cities; develop medicines; save lives. But this doesn't mean it is useful for you to think about it, only that it is useful to you that somebody else thought about it. It is not why I think about it, and not why I most want to tell this story.

You can get by perfectly well in life without understanding anything more about infinity than you did when you were five years old. But for me the usefulness of mathematics isn't about whether you need it to "get by" in life or not. It's about how mathematical thinking and mathematical investigation sheds light on our thought processes. It's about taking a step back from something to get a better overview. Flying higher up in the sky enables us to travel farther and faster.

Let's go.

Mathematics can be thought of as many things: a language, a tool, a game. It might not seem like a game when you're trying to do your homework or pass an exam, but for me one of the most exciting parts of doing research is when you're just starting something new and you get to play around with some ideas for fun. It's a bit like playing around with ingredients in the kitchen, which is more fun than trying to write down the recipe you invented in case you want to repeat it. And that is in turn more fun than trying to write down the recipe for someone else to be able to repeat.

I'm going to start by playing around with the idea of infinity a bit, to free our brains up and start exploring what we think might be true about it and what the consequences are. Mathematics is all about using logic to understand things, and we'll find that if we're not careful about exactly what we mean by "infinity," then logic will take us to some very strange places that we didn't intend to go. Mathematicians start by playing around with ideas to get a feel for what might be possible, good and bad. When Lego was first made, the designers must have played around with some prototypes first before settling on the wonderful final design.

A mathematical "toy" should be like Lego: strong enough to be able to build things, but versatile enough to open up many possibilities. If we come up with a prototype for infinity that causes something important to collapse, then we have to go back to the drawing board. After our initial games we'll be going back to the drawing board several times as we move through different ways of thinking about infinity that go wrong and cause things to collapse. When we finally get to something that holds up, it might not look the way you were expecting. And it causes some things to happen that you might not have been expecting either, like the weird fact that there are different sizes of infinity, so that some things are "more infinite" than others. This is a beautiful aspect of any kind of journey – discovering things you weren't expecting.

In the previous chapter I listed some beginning ideas about infinity.

Infinity goes on forever.

Does this mean infinity is a type of time, or space? A length?

Infinity is bigger than the biggest number.

Infinity is bigger than anything we can think of.

Now infinity seems to be a type of size. Or is it something more abstract: a number, which we can then use to measure time, space, length, size, and indeed anything we want? Our next thoughts seem to treat infinity as if it is in fact a number.

If you add one to infinity it's still infinity.

This is saying

∞ + 1 = ∞

which might seem like a very basic principle about infinity. If infinity is the biggest thing there is, then adding one can't make it any bigger. Or can it? What if we then subtract infinity from both sides? If we use some familiar rules of cancellation, this will just get rid of the infinity on each side, leaving

1   =   0

which is a disaster. Something has evidently gone wrong. The next thought makes more things go wrong:

If you add infinity to infinity it's still infinity.

This seems to be saying

∞ + ∞ = ∞

that is,

2 ∞ = ∞

and now if we divide both sides by infinity this might look like we can just cancel out the infinity on each side, leaving

2 = 1

which is another disaster. Maybe you can now guess that something terrible will happen if we think too hard about the last idea:

If you multiply infinity by infinity it's still infinity.

If we write this out we get

∞ × ∞ = ∞

and if we divide both sides by infinity, canceling out one infinity on each side, we get

∞ = 1

which is possibly the worst, most wrong outcome of them all. Infinity is supposed to be the biggest thing there is; it is definitely not supposed to be equal to something as small as 1.

What has gone wrong? The problem is that we have manipulated equations as if infinity were an ordinary number, without knowing if it is or not. One of the first things we're going to see in this book is what infinity isn't, and it definitely isn't an ordinary number. We are gradually going to work our way toward finding what type of "thing" it makes sense for infinity to be. This is a journey that took mathematicians thousands of years, involving some of the most important developments of mathematics: set theory and calculus, just for starters.

The moral of that story is that although the idea of infinity is quite easy to come up with, we have to be rather careful what we do with it, because weird things start happening. And that was just the beginning of the weird things that can happen. We're going to look at all sorts of weird things that happen with infinity, with infinite collections of things, hotels with infinite rooms, infinite pairs of socks, infinite paths, infinite cookies. Some weird things are like 1 = 0, not just weird, but undesirable. So we try to build our mathematical ideas to avoid those. But other weird things don't contradict logic, they just contradict normal life. Those weird things don't cause problems to our logic, they just cause problems to our imagination. But it can be very exhilarating to stretch our imagination just like fiction writers do when they create a person who lives infinitely long (immortality) or who can travel infinitely fast (teleportation). And it's not just exhilarating: it can shed new light on our normal life. When characters are immortal in fiction, they often end up realizing how the finiteness of life is actually what gives it meaning.

Infinite Hotels

When we start teaching children the idea of numbers, we usually give them some objects to think about, or we talk about numbers while they're eating something that comes in distinct pieces, like strawberries or beans. Or maybe we count spoonfuls of something as they eat it.

If we try to count spoonfuls of something until we get to infinity, we'll be here for a rather long time. Actually later on we are going to do something a lot like counting up to infinity, but for now instead of counting all the way there, we'll play with something that is already infinite: an infinite hotel.

Imagine a hotel with an infinite number of rooms, with room numbers 1, 2, 3, 4, … going on forever.

Now imagine that you're the manager of this amazing hotel, and every room is full. You're reveling in the amount of money you're raking in when another person arrives and asks for a room. On the one hand, the hotel is full. On the other hand, if you just ask everyone to move up a room …

This hotel with an infinite number of rooms is known as Hilbert's Hotel. It's named after the mathematician David Hilbert, who used it as a vivid way of illustrating the strange things that can happen when you start thinking about infinity. A normal hotel only has a finite number of rooms. Once it's full that's it – if another guest turns up, there's nothing you can do about it without building an extension. However, with the infinite hotel, you can get the person in room 1 to move to room 2, and the person in room 2 can move to room 3, and the person in room 3 can move to room 4, and so on. We can tell the person in "room n" to move to "room n + 1," and because we have an infinite number of rooms, every n does have an n + 1, so every hotel guest has a new room to move into. This leaves room number 1 empty, and the new guest can be accommodated.

This is sometimes called a paradox, but there's nothing wrong with the argument. It's just that the conclusion is counter-intuitive. How can we fit an extra guest into a hotel that's already full? The only reason this is counterintuitive is that we're too used to finite


  • "Ms. Cheng's chatty tone keeps things fresh. She has a knack for folksy analogies, and at different points in the book she illuminates different properties of infinity by discussing Legos, the iPod Shuffle, snorkeling, Battenberg cakes and Winnie-the-Pooh... she does a great service by showing us non-mathematician schlubs how real mathematical creativity works."
    --Wall Street Journal
  • "Our minds cannot truly grasp the concept of infinity, but Eugenia Cheng takes us on a wild journey to help us in our search for it. It's a small, unassuming symbol, but it holds a giant idea. Cheng helps us understand the basics of infinity and then takes us on a ride to see its most lofty applications. From the practical to the entirely theoretical, this is a book to watch for."
--Paste Magazine
  • "The idea of infinity is one of the most perplexing things in mathematics, and the most fun. Eugenia Cheng's Beyond Infinity is a spirited and friendly guide--appealingly down to earth about math that's extremely far out."—JordanEllenberg, author of How Not to Be Wrong and professor ofmathematics at University of Wisconsin-Madison
  • "Beyond Infinity is witty, charming, and crystal clear. Eugenia Cheng's enthusiasm and carefully chosen metaphors and analogies carry us effortlessly through the mathematical landscape of the infinite. A brilliant book!"—IanStewart, author of Calculating the Cosmos

    "[Cheng] conveys the spirit of inventiveness and creativity in math... Refreshing is the word that keeps coming to mind."—StevenStrogatz, as quoted in the New York Times
  • "Dr. Cheng...has a knack for brushing aside conventions and edicts, like so many pie crumbs from a cutting board."—NatalieAngier, New York Times
  • "[Eugenia Cheng's] tone is clear, clever and friendly. Even at her most whimsical she is rigorous and insightful... [She is] a lucid and nimble expositor."—AlexBellos, New York Times Book Review
  • On Sale
    Mar 28, 2017
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    Eugenia Cheng

    About the Author

    Eugenia Cheng is the scientist in residence at the School of the Art Institute of Chicago and an Honorary Fellow at the University of Sheffield. The author of How to Bake Piand Beyond Infinity, she lives in Chicago, Illinois.

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