Mark Vuletic
Infinity is both horrible and cool. This article will go over one way in which this is so.
Mathematicians call the non-fractional numbers starting with 1—that's 1, 2, 3, and so on—the natural numbers. There are infinitely many natural numbers. To say the same thing in a slightly different but useful way, the set (or collection) of all natural numbers is infinite.
Is it possible to add an object to an infinite set of objects? Sure it is. Take the set of all natural numbers, and then put a triangle in there with them. You have just add added an object to an infinite set of objects. Easy. Well, easy unless you are dealing with one of those poisonous triangles.1
Here's the question I want to pose to you: if you consider the two sets above—the set of all natural numbers, and the set of all natural numbers with a triangle thrown in—how do they compare? Does the first set contain more objects than the second? Does the second set contain more objects than the first set? Or do the two sets contain just as many objects?
The answer to this question may seem obvious to you. Of course the second set contains more objects than the first set, you will say. The second set contains all of the objects in the first set, and one more, so of course it contains more objects than the first set. Of course it does! Stop looking at me like that! Are you crazy? Have you gone mad? Is that what has happened? You've gone mad? You've lost your mind? Of course it contains more objects! Of course it does! Of course it does!
But the second set does not contain more objects than the first. It gets worse: the second set also does not contain fewer objects than the first. It gets even worse: the two sets also do not contain just as many objects as one another. But now it seems every possible answer to the question has been eliminated. This means that there must be something wrong with the whole question of how the sets compare, as inoffensive as the question sounds.
If you feel like tearing your hair out, I sympathize. As I said, infinity is horrible. My burden ahead will be to demonstrate to you that the claims above are true. If I succeed, you may still feel like tearing your hair out—I know I still do (my hair, not yours)—but hopefully you will also feel some awe mixed in with all of that, because it is amazingly cool that one can prove something so seemingly absurd.4,5
Let's do a couple of things with labeling to make the discussion ahead easier to follow.
First, let's name the two sets we want to compare. The first set contains all of the natural numbers and nothing else, so let's call that set NUM (for Numbers). The second set contains all of the natural numbers and a triangle, so let's call that set NUMTRI (for Numbers and Triangle).
Second, let's relabel the numbers in the two sets. We will attach an "n" to the front of the numbers in NUM, and attach a "t" to the front of the numbers in NUMTRI. To remember which letter goes with the numbers of which set, just remember that "NUMTRI" contains a "t" and NUM does not. So:
1, 2, 3, and so on in NUM get relabeled as n1,n2, n3, and so on.
1, 2, 3, and so on in NUMTRI get relabeled as t1, t2, t3, and so on.
The triangle in NUMTRI just remains as it is. Since there's no triangle in NUM, there's no need to relabel the triangle in NUMTRI.
I trust you will agree that mere relabeling does not change the number of objects in each set.
The advantage of all of the labeling and relabeling we have done above is that now we get to say things like "Pair off n1 with t1" instead of "Pair off 1 in the set containing only the natural numbers with 1 in the set containing the natural numbers and the triangle." In a paragraph full of statements like these, it will be much easier to keep track of what's going on if we use the abbreviations.
Now, let's get to the argument.
An aside about counting
Sometimes we compare sets by counting the objects in them and then comparing the numbers. But this actually is a way of pairing off elements in different sets: the only difference is that a third set is brought into play as an intermediary. When you count objects in a set, you pair each object in the set with a natural number, starting with 1 and then proceeding one-by-one to higher numbers. Then you let the set of numbers stand as a proxy for the original set. So, when you count up two sets, you end up with two proxies, and you can compare their sizes. The chief advantage of counting is that the way we set up the counting procedure (start with 1, match the next with 2, and so on) guarantees that the largest number in the number set will also tell us the number of objects in the set. But it all still comes down to pairing off objects.
Here is the probable reason why you think NUMTRI contains more objects than NUM. When you compare two sets, you probably do it like this: you pair off the objects in one of the sets with the objects in the other, and then look to see whether any objects are left over. If there are no leftovers, then the sets have just as many elements. If there are leftovers, then the set that has leftovers contains more elements. For instance, if we want to compare the set of students in a course with the set of chairs in a classroom, one obvious way to do this would be to try to seat every student: this is a way of pairing off students with chairs. If no students are left standing and no chairs are left empty, then there are just as many students as chairs. If any student is left standing, then there are more students than chairs. If any chair is left empty, then there are more chairs than students. If there are students left standing and chairs left empty, then you have a serious discipline problem with your students. (If you are reading all of this and saying, "No, no, no—that's not how I would do it. I would just count the chairs and students and compare the numbers," then see the sidebox about counting.)
So, when we compared NUM with NUMTRI above, here's what we did: we paired off n1 with t1, n2 with t2, n3 with t3, and so on. When we paired the elements off this way, every single number in each of the sets got paired off. So, each object in NUM was matched to an object in NUMTRI and each of the non-triangle objects in NUMTRI was matched to an object in NUM. But this meant the triangle in NUMTRI was left over. So, we concluded—understandably enough—that NUMTRI contains more elements than NUM.
The problem with this procedure, as natural as it feels, is that it does not work for infinite sets. It works just great for finite sets, and feels natural to us because most of the sets we deal with are finite. But the procedure breaks down when you try to apply it to infinite sets.
The thing is, there is no rule that says we have to pair off n1 with t1, n2 with t2, n3 with t3, and so on. In our student-chair example, for instance, you can seat the students however you want, as long as you don't (1) try to cram more than one student into a single chair, (2) let a student stretch out across more than one chair, or (3) leave a chair open when a student is still standing. You can seat the students alphabetically, according to height, according to who likes chinchillas more, according to who chinchillas like more, according to who are more like chinchillas, or whatever. It makes no difference.
So let's try out a different way of pairing off the objects in NUM with the objects in NUMTRI. The very first thing we do is pair off n1 with the triangle. Then we pair off the remaining objects, all at once, in the following fashion: n2 gets paired off with t1, n3 gets paired off with t2, n4 gets paired off with t3, and so on. When we do this, we find that everything in both sets gets paired off: there are no leftovers in either set. So, it now seems that the two sets contain just as many elements as one another.
This result probably feels wrong to you, but I haven't tricked you. If I choose any object out of NUM, or any object out of NUMTRI, you can tell me exactly which object it is paired off with. The fact that you can do that shows that all of the objects in both sets are paired off. Any uneasiness you may feel—any feeling, for instance, that there has to be something left over in NUMTRI—almost certainly comes from being accustomed to working with finite sets. But don't trust your feelings over what you have shown. Infinite sets don't have to obey our intuitions. And why should they?
We're not quite finished showing how horrible (and cool) the situation is. We have shown that you can pair off the objects in NUM and NUMTRI so that NUM appears to contain fewer objects than NUMTRI, and that you can pair off the same objects so that NUM appears to contain just as many objects as NUMTRI. But we can also pair off the same objects so that NUM appears to contain more objects than NUMTRI. There is nothing to stop us from pairing off n1 with the triangle, and then pairing off n2 with t1, n4 with t2, n6 with t3, and so on. When we do this, every element of NUMTRI is paired off with an element of NUM, but there are leftovers in NUM. In fact, all of the odd-numbered elements in NUM except for n1 are left over. Now it seems that NUM contains more—infinitely more—elements than NUMTRI.
We have now seen that if we try to compare NUM and NUMTRI in the normal way, then all hell breaks loose. Depending on how we chooses to pair off the objects in the two sets, we can make NUM contain fewer objects than NUMTRI, just as many objects as NUMRTI, or more objects than NUMTRI. Thus, it would be false to say that NUM contains fewer objects than NUMTRI. Likewise, it would be false to say that NUM contains just as many objects as NUMTRI or that NUM contains more objects than NUMTRI. The whole notion that NUM and NUMTRI compare at all in the ordinary sense, is mistaken.
This still is horrible, but hopefully now it seems horrible in a mind-blowingly awesome sort of way.
Since I said that there is something wrong with the notion that NUM and NUMTRI compare "in the ordinary sense," you may wonder whether there is an unordinary sense in which they do compare. This is mostly a topic for another day—hence, the "Part I" in this article's title—but I want to give you a prelude. Mathematicians do compare infinite sets, but only using the notion of cardinality. Mathematicians say that two sets have the same cardinality if the objects they contain can be paired up so that there are no leftovers in either set. Whether or not two sets can be paired up in some other way that does leave leftovers is not relevant to the notion of cardinality, so you can see here a difference from our normal sense of how sets compare. You probably also can see that NUM and NUMTRI have the same cardinality, because one of the ways of pairing off the objects they contain—the second way above—had no leftovers. Mathematicians designate the cardinality of sets like NUM and NUMTRI with the cool symbol \( \aleph_0 \) (or at least cool if you don't normally write in Hebrew).6
Are there infinite sets of different cardinality than \( \aleph_0 \)? That's the topic for next time.
1 Haven't you ever head of acute poisoning? AHAHAHAHA!2
2 Yes, this is the kind of thing you can expect from this article. If you prefer your discussions of infinity to be dry and sober, you can go read a textbook on set theory instead.3
3 I know what you're thinking: footnotes to footnotes? Allow me to address your concern: yes, footnotes to footnotes.
4 You seriously need to chill out, by the way. What was up with your rant two paragraphs ago? Who talks like that?
5 If you are wondering what happened to footnotes 2 and 3, look at footnote 1.
6 \( \aleph_0 \) reads "aleph nought," by the way.7
7 What, I can't write a serious footnote for a change?
Last updated: 27 July 2015
Copyright © 2017, Mark Vuletic. All rights reserved.