Introduction

What is infinity? As children, we are told that infinity is the number of things bigger than any other number of things. It is a one size fits all measure of things that are unfathomably huge. Because infinity is not something anyone will ever encounter unintentionally during his or her academic career, this notion of infinity remains with us throughout our lives.

In this series of posts, I want to look at some interesting and surprising details of infinities (wait… there are more than 1?). I will do this in 3 parts. In Part 1 (this part), I will introduce the notion of one-to-one correspondences, an important fundamental idea underlying the discussions that follow in parts 2 and 3. In Part 2, we encounter the ideas of “countable” vs. “uncountable” infinities. We will look at what each is, how they differ, and in doing so encounter one of the most elegant, ingenious mathematical proofs ever conceived, known as Cantor’s diagonalization argument. Finally, in Part 3, I discuss an interesting paradox that infinity brings about. I think that it’s worth considering one of the ways infinity shatters intuitions shaped by common, everyday encounters we have with our finite-in-every-way world.

As a note before you continue reading, this series of posts will be slightly more technical in nature than I’d like. There aren’t many day-to-day examples of infinity, so the examples I give will naturally be a little more abstract. That said, I’ll do my best to present these ideas in a fun, lucid and relatable way. My request to you the reader is that you allow infinity to bend your mind a little further than you might otherwise be comfortable with. It’ll be worth your while.

One-to-one correspondences: necessary background

Before we try to grapple with the infinite, we have to understand what it means for two sets of things to have the same size. Imagine, for instance, that we want to know if a set of people will fit in a set of theater seats. One way to figure this out is to count the number of things in each set (the people and then the seats) and see if the totals match. But that seems like too much work; to know if there are as many seats as there are people, I just need to figure out whether or not each person can be matched with exactly one seat. To do this, I can simply ask the moviegoers to sit down. If there are any people left standing or seats not filled, we know that the sets (of people and seats) have different sizes. This idea of matching one element of one set to exactly one element of the other is called creating a one-to-one correspondence between those sets. Sets have the same size only if you can show a one-to-one correspondence between them.

When we think about finite sets, we can always put a concrete number on the number of things in each one; there are 12 oranges in the bowl, 100 cars in the parking lot, and 7 billion people on earth. As soon as we consider infinite sets, this ingrained measure of size fails. The one-to-one correspondence is a tool that will allow us to compare the sizes of infinite sets in the coming discussion.