Why Infinity Is No Ordinary Number
The idea of infinity is easy to come up with, but we must be careful what we do with it.
The following is an excerpt from Beyond Infinity: An Expedition to the Outer Limits of Mathematics, by Eugenia Cheng
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
∞ + ∞ = ∞
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
∞ x ∞ = ∞
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.
Excerpted from Beyond Infinity: An Expedition to the Outer Limits of Mathematics, by Eugenia Cheng. Copyright 2017. Available from Basic Books, an imprint of Perseus Books, a division of PBG Publishing, LLC, a subsidiary of Hachette Book Group, Inc.
Dr. Eugenia Cheng is Scientist in Residence at the School of the Art Institute of Chicago and author of several books, including The Art of Logic in an Illogical World (Basic Books, 2018) and Is Math Real?: How Simple Questions Lead Us to Mathematics’ Deepest Truths (Basic Books, 2023).