To Infinity and Beyond: an In-Class Exercise

One of the problems I encounter when I am doing outreach work at high schools is the remarkable lack of confidence that some teachers have in their own students. I’ve decided today to post one of the activities I do for outreach with comments on the teacher’s reactions. This activity arises from an online argument that occurs fairly often in which someone wants to know how on earth we know that infinity exists. The answer to this question is usually entirely unsatisfactory to the person that asked it, but it does contain an important cultural fact about math.

Mathematicians do not know if infinity exists.

Can you see now why this answer is potentially infuriating? All mathematics rests on a foundation of unproven assumptions called axioms. One of the things that makes Euclid’s Elements a remarkable work is that he did an great job of choosing a small and reasonable set of axioms for plane geometry. Modern math is (mostly) founded on the Zermelo-Fraenkel axioms plus one other axiom, the axiom of choice, that causes a lot of trouble but which seems to be needed. The seventh axiom of the Zermelo-Fraenkel list says “An infinite set exists.” This means that math assumes the existence of infinity and works onward from that assumption.

The in-class exercise

One thing we worry about in mathematics is how to represent numbers with as few symbols as possible. A set is a list of distinct objects between curly braces so that “{}”, for example, is the empty set. If we want to represent the counting numbers then we can do it with sets that have as many members as the number we are representing. This seems pretty weird but this gives us the following (long) ways of writing numbers with very few symbols:

Number As a set of numbers Full set crazy
0 {} {}
1 {0} {{}}
2 {0,1} {{},{{}}}
3 {0,1,2} {{{},{{}}},{{},{{}}}}
4 {0,1,2,3} {{{{},{{}}},{{},{{}}}},{{{},{{}}},{{},{{}}}}}

If we needed to use the numbers for something like counting change, this would be insane. So why bother? Well, what number is this?

{0,1,2,3,…}

At this point in the exercise, I wait patiently for a student to answer. On several occasions when I do this exercise, I’ve had the teacher signaling from the back of the room that I am out of bounds, doing something so weird it will damage the students, or that I should throttle back. I’m not sure exactly what the signal means (because it is non-verbal). In defense of teachers in general, my second biggest problem with this demonstration is the teacher getting real interested and blurting out the answer I want. Almost always someone, a student or the teacher, says:

That’s infinity!? And it is. In this laborious way of writing numbers, all the counting numbers smaller than the number your currently writing out are members of the set. In the table above, for example, 0, 1, 2, and 3 were all members of 4. That means that the set of all the counting numbers is bigger than every counting number – in other words is bigger than any finite number – and so it is reasonable to conclude that the number {0,1,2,3,…} is (some kind of) infinity. In the title I said “to infinity and beyond“, so now I need to pay up. Look at the following:

4={0,1,2,3}={0,1,2} ∪ {3}=3 ∪ {3}The symbol is “union” which combines the members of two sets. This example shows us that n+1=n ∪ {n} – in English, a number plus one is the union of that number with the set of that number. A little weird, but that also means that if w={0,1,2,3,…} then w∪{w}={w,0,1,2,3,…}=w+1, and we just found a way to write down infinity plus one. Infinity plus one is the set of all the finite counting numbers together with the infinite number we already found. You can repeat this trick to get infinity plus 2 or 3 or whatever. There is a whole branch of mathematics that studies different types of infinities, but that is well beyond what I want to cover in this post.

A couple of times, I’ve really enjoyed the expression on the face of a teacher when one of their students says “that’s infinity,” but there is still the problem that, if you lower your expectations, people will tend to meet them. Having high expectations of your students and then also being generous with help is a good receipt for teaching. To be fair, some teachers are not given the resources – especially time – needed to make this work. In Ontario now we are are having a teacher’s strike where one of the big issues is class size. Class size is strongly related to the amount of time a teacher has for planning and for helping their students.

The really weird way of writing down numbers used in this in-class exercise is not a useful way to represent numbers for most purposes, but it did give us access to simple arithmetic on infinite numbers. This is a good example of a general principle:

A powerful mathematical tool is to change your point of view until the problem becomes easy.The funny way of writing numbers is a different point of view – it makes a lot of things much harder but it makes infinite arithmetic, if not easy, at least possible. This post, while it does touch the Occupy Math theme concerning math anxiety, is a good deal more abstract than our typical post. Please comment or tweet to tell us if you would like, or don’t mind skipping over, occasional posts like this.

I hope to see you here again,
Dan Ashlock
Department of Mathematics and Statistics
University of Guelph, Ontario, Canada

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