Roots of Unity

Roots of Unity

Mathematics: learning it, doing it, celebrating it.

Some Infinities Are Bigger than Other Infinities, and Some Are Just the Same Size


Warning: contains minor spoilers for The Fault in Our Stars.

I recently read The Fault in Our Stars by John Green, now a major motion picture that has led to theft in Amsterdam and a shortage of dry eyes in movie theaters around the world. One of the ideas that resonates with Hazel, the 16-year-old narrator of the story, is the idea that "some infinities are bigger than other infinities."

In Hazel's voice, Green writes,

“There are infinite numbers between 0 and 1. There’s .1 and .12 and .112 and an infinite collection of others. Of course, there is a bigger infinite set of numbers between 0 and 2, or between 0 and a million. Some infinities are bigger than other infinities.… I cannot tell you how grateful I am for our little infinity. You gave me forever within the numbered days, and I’m grateful.”

The sentiment is lovely but mathematically inaccurate. One of the most mind-blowing facts a young mathematician learns is that, in a specific, rigorous way, there are exactly as many numbers between 0 and 1 as there are between 0 and 2, 0 and a million, or even in the entire set of real numbers! Don't worry, it's natural to feel dubious about that. It seems impossible that a set could be the “same size” as a set that contains it plus some other stuff! But that’s one of the marvelous mysteries of infinity.

How to count potatoes by pairing them with numbers. Image: Yen Duong.

Essentially, the way to tell whether two sets are the same size is to see whether you can pair up elements so you use all the elements in each set exactly once. Georg Cantor, whom Green references earlier in the book, proved that there are indeed different sizes of infinity. But the infinities between 0 and 1 and 0 and 2 are not different sizes. Each number between 0 and 1 can be doubled to get a number between 0 and 2, and each number between 0 and 2 can be halved to get a number between 0 and 1.

My math blogging pal Yen Duong of Baking and Math just wrote a post about this mathematical fault in The Fault in Our Stars that explains Cantor’s diagonalization argument with adorable cartoons of potatoes, so you can check that out for more details. Mathemusician Vi Hart also made a beautiful video about the ideas. It doesn't have any potatoes, but it does have lots of clouds.

Although I make a living by being a pedantic math teacher who tortures gently encourages students to be precise and rigorous, the mathematical error in this novel doesn’t bother me. I know it bothers some other people, especially given Green's role as the host of the Mental Floss video channel, and I can understand why, but I don't feel the same way.

I don't mind it if a teenager in a book (or, for that matter, in real life) doesn't understand Cantor’s diagonalization argument but still finds the idea of bigger and smaller infinities meaningful. I don’t know whether Green intended for Hazel’s understanding of infinite cardinalities to be accurate or not, but if you are familiar with the fault in her argument, I think it lets you read the passage in a different way. Hazel and Augustus are both smart, thoughtful kids who are coping with terrible circumstances, but they also have that combination of naiveté and pretentiousness that strikes me as quintessentially adolescent, and I see Hazel's misunderstanding of Cantor as highlighting those attributes. Whether Green intended it or not, that's how I read it.

That said, I think Hart's video has a more beautiful interpretation of infinity as it applies to star-crossed lovers’ too-short lives. (Incidentally, Green credits Hart with helping him think and write about some essential themes in the book.) Around the 9:25 mark of the video, after explaining Cantor's diagonalization argument and some of the different infinities we know about, she puts it perfectly:

“Whether those different sorts of infinities apply to something like moments of time is unknown. What we do know is that if life has infinite moments, or infinite love, or infinite being, then a life twice as long still has exactly the same amount. Some infinities only look bigger than other infinities. And some infinities that seem very small are worth just as much as infinities ten times their size.”


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