(written at some point in 2015)

When I took my first upper level math class, I learned about Sets. In principle, sets are simple. Sets are collections of things. Sets can be big or sets can be small. Sets can contain people. Sets can contain musicians, molecules, and animals. However, in math class, it was appropriate that most of our sets comprised numbers.

And oh, how they did. We worked with sets containing numbers of all shapes and sizes. I saw beautiful sets and ugly sets. I saw sets defined by patterns and sets defined by other sets. I saw sets that contained nothing, sets that contained each other, and sets that contained themselves (see: Russell’s Paradox). I saw sets I never could’ve dreamed of before.

In 1874, German mathematician Georg Cantor, the inventor of set theory, explained the difference between “countable” and “uncountable” sets. (He wrote in German, but the terminology is close enough.) Take \(\mathbb{N}\), the set of “Natural Numbers.” This is the set of whole, non-negative numbers, that is,

\[\mathbb{N} = \{0, 1, 2, ... \}\]

The number of elements in $\mathbb{N}$ is called its “cardinality.” And, every set with equal cardinality to $\mathbb{N}$ is called a “countable” set.

Alternatively, every set with a greater cardinality than $\mathbb{N}$ is known as “uncountable”. This is intuitive, seeing as these sets by definition contain too many items to be “countable.” An example of an uncountable set is the set of non-negative real numbers, or the set:

\[\mathbb{R}^{+} = \{ x \in \mathbb{R} : x \ge 0 \}\]

There’s no way to put these numbers in a list. Pick zero as the first number. What number comes after it? 1? 0.1? 0.0001? Whichever number you choose, and there will always be a smaller one to take its place. Try as you might, there is no pattern that allows you to N all these numbers in one sequence, giving each one a definite index. This is why we call this set “uncountable.” (A weirder example is the set of rational numbers, numbers that can be defined as the ratio of two integers. It turns that this set is, in fact, countable. See: Cantor’s Diagonal Argument.)

Mathematically, I was fascinated by “countable” sets, but I struggled to accept the terminology. By my reasoning, a countable set isn’t countable at all. If a set is countable, it’s countable, and if it’s infinite, it’s infinite. Cantor gave countable sets this name because they can be ordered in such a way such that they, eventually, could be counted. For example, with the natural numbers, you could start with zero and add one for each successive element. Theoretically, in this pattern, the set is “countable.”

But that’s not what countable means. Here’s the paradox: you could start at zero, spend a lifetime adding ones, and never reach the end of the set. In fact, you could count until the end of time and you wouldn’t even be getting closer to the end of the natural numbers. That’s because there is no end. For this reason, some mathematicians prefer to call these sets “countably infinite.”

The notion that something can be countably infinite isn’t just a paradox, it’s a fallacy. It suggests a task that can never be completed. And this is a depressing way to look at life. It’s as if somewhere a bunch of mathematicians locked a child in a room and told him that he wasn’t allowed to come out until he finished counting. But there are an infinite amount of natural numbers. He argued. Countably infinite, they corrected. And he’s still counting.

We’ve stuck with Cantor’s terms since he coined them, even though they represent a sense of false hope for novice mathematicians everywhere. However, it doesn’t have to be this way. I suggest the use of the phrase “infinitely indexable” instead of “infinitely countable” for these types of sets. This relays the fact that each element of the set has a specific position in between two other elements of the set. But this doesn’t misrepresent the elements as “countable.” Because they’re not. And they never will be.

They’ll never be countable, says the boy in the room. But they are indexable. Just start at zero.

The door unlocks.