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

The views expressed are those of the author and are not necessarily those of Scientific American.

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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.”

Evelyn Lamb About the Author: Evelyn Lamb is a postdoc at the University of Utah. She writes about mathematics and other cool stuff. Follow on Twitter @evelynjlamb.

The views expressed are those of the author and are not necessarily those of Scientific American.

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  1. 1. Thalesdisciple 3:19 pm 07/10/2014

    John Green was, in fact, aware that Hazel’s interpretation of this mathematical statement is inaccurate:

    This fall is going to be the semester of John Green in my classes. Literally the first topic in analysis is “some infinities are bigger than other infinities”. And in pre-calculus, after reviewing some of the basic types of functions, I’m going to have the students study the formula from An Abundance of Katherines:

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  2. 2. lavaroy 4:06 pm 07/10/2014

    Dear Ms. Lamb, do you feel (I do) that infinity does not map to the physical universe? Dividing the number line infinitely many times runs afoul of quantum physics, which has a smallest possible divisor. Infinity was once called a dangerous idea. Pondering infinity drove Cantor mad. Infinity, and its inverse, singularity, in the hands of physicists, gives us the unfathomable multiverse. To consider infinity is to ponder the imponderable. I think Cantor’s argument proved that infinity contradicts itself. Infinity is non-mathematical. You can’t have an equals sign and an infinity sign on the same page. I think infinity should be taken out of the hands of mathematicians and laid in the hands of poets like Hazel.

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  3. 3. Evelyn Lamb in reply to Evelyn Lamb 4:15 pm 07/10/2014

    I do believe that the physical universe has constraints that mean that there is a limit to how large or small something that we experience can be. But I work in the mathematical world, where circles are perfectly circular and lines perfectly one-dimensional, not made up of all those messy atoms that jiggle around. In fact, I think infinity is a perfect mathematical concept, it just doesn’t translate into the physical world. Cantor did have problems, but there are plenty of people who have pondered infinity without having mental breakdowns.
    I’m glad that both poets and mathematicians have their hands on infinity!

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  4. 4. John-sensei 9:15 pm 07/10/2014

    If your measure of a set is “cardinality”, you are correct. However, there are other measures, and other ways to compare sets. I was recently talking to a mathematician, and he was saying something like “A is uniquely the largest set with property X”. “Wait a minute”, I said. “B has the same cardinality as A and satisfies X.” “I didn’t mean cardinality” he replied. “B is smaller than A in the sense that it is a proper subset of A.” The interval from 0 to 1 is smaller than the interval from 0 to 2 in precisely this sense.

    Consider the way we measure things in calculus. Calculus depends on the infinite division of continua using methods that keep track of the fact that even after such an infinite division there’s twice as much stuff in [0,2] as there is in [0,1].

    And then there’s renormalization…

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  5. 5. Petros1513 12:30 am 07/11/2014

    Great article Evelyn.

    I wanted to build on the previous comment and ask what you thought about the ontological status of mathematics and its relationship with the world (I know this is a bit off topic, apologies for that). I know a majority of mathematicians, as well as many philosophers and physicists, believe that mathematics is in some sense real and discovered rather than invented by humans. I’m strongly in agreement with that sentiment, and the fact that mathematical relationships are present everywhere in the world (very abstract symmetry groups even predict particles that have later been discovered in particle accelerators) lends strong credence to that view in my opinion.

    Just wanted to get your thoughts on the subject. I always like checking in to this blog and of course I know you don’t focus on philosophy of mathematics, but I figured I’d ask what you thought.

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  6. 6. lpwagner 2:59 am 07/11/2014

    strictly speaking, mathematicians never really use infinity. The infinity symbol is, technically speaking, a shorthand that means something specific in whatever context one is using it in or is defined as numbers filling the gap at the extremes of the real numbers. Much of Real Analysis is an attempt to talk about infinity without actually talking about it. For instance, the symbols “lim x->infinity of f(x)=A” is shorthand for “for every e>0, there is an M>0 such that if x>M, then |f(x)-A|<e." We just use the infinity symbol to shorten our notation.

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  7. 7. Evelyn Lamb in reply to Evelyn Lamb 3:10 am 07/11/2014

    Indeed this is not a philosophy of mathematics blog, and I have not had any formal training in philosophy, nor do I spend much time thinking about it in my day-to-day life as a mathematician. But that never stopped anyone on the internet from saying their opinions! I think mathematics is a wonderful blend of creation and discovery. Human invention or creation is an important part of mathematics, but it’s amazing what we can discover lurking there after we set up our definitions and axioms. Mathematics does have amazing correspondences with real-world phenomena, but the mathematical phenomenon is always an abstraction of the physical phenomenon, in a simplified, idealized case. It truly is an unreasonably effective way of studying physical reality, as an article by Eugene Wigner puts it.
    But mathematics doesn’t have to be an abstraction of a physical system. I think we tend to prefer to do mathematics that seems to have some basis in reality, but we could set up any definitions and axioms we wanted at the beginning and still be doing mathematics. We just tend to set up definitions and axioms that seem to relate to things we’ve seen in the real world.

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  8. 8. Evelyn Lamb in reply to Evelyn Lamb 3:19 am 07/11/2014

    John-sensei: you do need to define what you mean by “large” and “measure,” and different definitions make sense for different purposes. When mathematicians talk about bigger and smaller infinities, we’re always talking about the sense of cardinality. But when we talk about sets, we might use different definitions of “larger,” as in the example you shared. This comes up in my analysis classes: the interior of a set A is the “largest” open set contained in A, in the sense that every open set contained in A is a subset of the interior of A.
    Of course, there is a very natural way of measuring the intervals [0,1] and [0,2] that makes the second one bigger than the first, namely length on the real line, but both intervals are finite in that measure.

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  9. 9. jtdwyer 6:48 am 07/11/2014

    Nice article – although I don’t do math. However (to pick a nit), since determining that the number of elements in a set is infinite implies that the number cannot be _precisely_ determined – isn’t it incorrect to say:
    “… 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?”

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  10. 10. Evelyn Lamb in reply to Evelyn Lamb 7:52 am 07/11/2014

    The cardinality of an infinite set can often be precisely determined (as it can in these cases), and that’s the “specific, rigorous way” that I’m referring to in that sentence. It is precise to say that the cardinality of the natural numbers, or the rational numbers, or the primes, is aleph null. It’s also precise to say the cardinality of numbers between 0 and 1 is the same as the cardinality between 0 and 2. It feels different than saying, “I ate 13 gummy bears,” but it’s just as precise.

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  11. 11. John-sensei 4:03 pm 07/11/2014

    If you want to be pedantic about it, the quote you posted includes “Of course, there is a bigger infinite set of numbers between 0 and 2”. To me, that establishes that Hazel is ordering infinities by the rule that a set is bigger than its proper subsets.

    You should not mislead students by teaching that cardinality is *the* measure of infinity. It is a conventional measure, but, as we’ve established here, the convention depends on context. In my first algebra class, I learned that a quadratic equation has zero, one, or two solutions. Later, I learned that the Fundamental Theorem of Algebra demands two. Which is it? Context matters.

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  12. 12. Edgod1 2:27 am 07/12/2014

    I think it has significance with judges in courts. For example, if someone has an only child and someone else kills that child, the feeling of loss may be infinite. But someone else may have had 20 children, and someone murdered 10 of them. The judge gives the murderer a sentence of 50 years imprisonment. This is then used as a precedent in the first case, the judge saying that killing one child should only merit the sentence of giving one tenth of the sentence a person gets for killing 10 children. But it could also be argued that losing half one’s children is half as bad as losing all one’s children. As humans, we put a value on things, which has a psychological aspect as well as a mathematical one. How much worse is losing 10 children than losing one child? This is like trying to multiply infinities, do you still get infinity or 10 x infinity? Is there a difference, and if so is it always or just sometimes?

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  13. 13. didomenica 1:13 pm 07/12/2014

    Infinity and zero are two sides of the same coin! see:

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  14. 14. Evelyn Lamb in reply to Evelyn Lamb 4:05 pm 07/12/2014

    John-sensei: It is *possible* that that is what Hazel meant, but I think the book implies that the context is cardinality. Earlier, when they learn that there are bigger and smaller infinities, the person who tells them about it says, “no one really solved it until Cantor showed that some infinities are bigger than other infinities.” So it’s pretty clear that Cantor’s diagonalization argument is what’s being referred to. I didn’t include that in the original post because I didn’t want to be that pedantic about it, but since you offered… :)

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  15. 15. christinaak 3:30 pm 07/13/2014

    In some cyclic models (including the Evolutionary Cyclic Model) it is infinite space that expands and contracts due to thermodynamic instabilities inherent within the universe. Whether the universe is expanding (as it is now) or contracting (as it will at the termination of the current cosmic cycle), it is always infinite in extent.

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  16. 16. lavaroy 4:29 pm 07/15/2014

    Hazel’s expression, “…our little infinity, within the numbered days…”
    If time is a one-dimensional ray, her infinite set (love) must exist radially to the ray of time. All forms of energy include time, but extend into spatial dimensions. Maybe the number line is like this? One dimensional when you travel on it, but strongly curved into waves (sets) when viewed from a distance. Waves on the number line seems like a legitimate way to picture “different size infinities.” Maybe?

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