Counting all the numbers between zero and one

Tony Pisculli

2 May 2019

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Heavy as the setting sun
Oh, I’m counting all the numbers
between zero and one
Happy, but a little lost
Well, I don’t know what I don’t know
so I’ll kick my shoes off and run

Sir Sly, &Run

There’s a song popular on the radio right now (‘&Run’ by Sir Sly) that contains the lyric, “I’m counting all the numbers between zero and one.” The first time I heard it, I thought, that’s not going to take very long. I assumed the singer was referring to integers, of which there are none strictly between zero and one. Was he trying to tell the woman he was singing to that he was impatient to see her again? Or suggesting that they were as close as two adjacent integers, with literally nothing keeping them apart?

The second time I heard the song, though, it occurred to me that he might be referring to the real numbers, of which there are an infinite number between zero and one. Counting those would keep him busy for, well, ever. Was he expressing his (infinite) patience? Or was he in despair at an impossible task? It all depends on what Sir Sly means by ‘numbers.’ (Also ‘counting’, and even ‘between’.)

The music of set theory

Natural languages, like English, are rife with these sorts of ambiguities. Sometimes they lead to disastrous misunderstandings. Other times, especially in poetry, jokes and songs, they’re used for deliberate effect. Mathematicians, on the other hand, aim for clarity, employing special notation that can be daunting if you’re not familiar with it but is more concise and less prone to misinterpretation.

Consider ‘between’. Do all the numbers between zero and one include zero and one? It depends on context. If we’re asked, what number is between two and four, then three is an acceptable answer but two and four are not. On the other hand, if we’re asked to choose a number between one and ten, then one and ten are valid. A game that’s billed as “fun for kids between eight and eighteen” probably doesn’t mean to exclude eight-year-olds and eighteen-year-olds.

With mathematics, we can be more precise, using set builder notation to specify exactly what we mean. For example, let’s define the set A, to capture one interpretation of the song lyric, as

$$A = \{ x |~x \in \mathbb{Z}, 0 < x < 1 \}$$

which you can read as “A is the set of all x such that x is an element of the integers, and x is greater than zero and less than one.” Or, more informally, “all the integers strictly between zero and one,” which is getting closer to what Sir Sly said but doesn’t scan as well. (Note that $\mathbb{Z}$ is the set of all integers, z being the first letter of zahl, which means number in German.)

Another interpretation of the numbers Sir Sly is counting is

$$B = \{ x |~x \in \mathbb{Z}, 0 \leq x \leq 1 \}$$

which is almost identical to A but includes zero and one.

We can create two more sets, C and D, to cover the case where Sir Sly is referring to real numbers:

$$C = \{ x |~x \in \mathbb{R}, 0 < x < 1 \}, \\
D = \{ x |~x \in \mathbb{R}, 0 \leq x \leq 1 \}$$

Then C corresponds to the interpretation where Sir Sly is counting all the real numbers strictly between zero and one, and D corresponds to the interpretation where they’re counting all the real numbers between zero and one inclusive. (Note that $\mathbb{R}$ is the set of all real numbers, and, yes, r stands for real.)

By the way, C and D can be expressed even more compactly as the intervals $(0, 1)$ and $[0, 1]$, respectively. An interval is just a contiguous subset of the real number line. Square brackets mean that the endpoints are included. Parentheses, or round brackets, mean the endpoints are excluded.

Let’s count

So far, then, we have four different interpretations of what “the numbers between zero and one” could be. Let’s look at our last ambiguous word, ‘counting’, which has multiple conflicting meanings in English. In mathematics, once again, there is a single meaning which is clear and specific. To count the elements of a set, we put them in one-to-one correspondence with a subset of the naturals, $\mathbb{N} = \{ 1, 2, 3 \ldots \}$. For example, suppose we have a set of elephants:

$$E = \{\mbox{Dumbo, Jumbo, Babar, Horton, Tyke}\}.$$

To count this set, we put them in correspondence with the naturals, like so:

The ambiguity with ‘counting’ in English comes from the distinction between simply determining the count and performing the count (i.e. counting each elephant, one by one, perhaps out loud). The first is instantaneous; the second takes time. If you’re counting the number of eggs in ten cartons of a dozen each, you can simply inspect them to see if any are missing. If all eggs are present, the count is 120. You don’t have to count them individually. On the other hand, if you’re giving someone a head start in hide-and-seek, you’re expected to actually count each number individually — to yourself or out loud, with alligators, possibly — and not just shout “one hundred!” before you start looking.

In mathematics, the first definition applies and counting takes no time. For the purpose of interpreting the song, it’s probably fair to assume that Sir Sly intends the second meaning and is performing the count.

Some Sly counting

So, what does it mean to count all the numbers between zero and one for our various interpretations, $A$, $B$, $C$ and $D$?

For set $A$ there’s nothing to do. $A$ is the empty set; there’s nothing to count.

For set $B$, counting the elements means putting them in one-to-one correspondence with the naturals, e.g.

Metaphorically there’s not a lot to distinguish $A$ from $B$ in terms of effort, unless Sir Sly counts very slowly. Even counting by alligators (one-alligator, two-alligator), that won’t take long.

When we get to set $C$, however, we run into trouble. $C$ is literally uncountable. The interval $(0, 1)$ cannot be put into one-to-one correspondence with the naturals.

But wait — the set of natural numbers is infinite. Surely that should be sufficient to count the infinite reals?
You might think so. For instance, it’s possible to put the integers into a one-to-one correspondence with the naturals, even though, at first glance, there seem to be twice as many of them.

A naive first attempt, lining up naturals with the integers beginning at zero, fails:

We’re left with all the negative numbers unaccounted for, like assembling an Ikea bookshelf and being left with a troubling extra part, only worse since there’s an infinite amount of them. There’s no way to deal with all the positive integers (and zero) first and then somehow get around to the negative integers later. There is no later, since the process continues forever.

But if we rearrange the integers, like so:

$$\mathbb{Z} = \{ 0, 1, -1, 2, -2, 3, -3 \ldots \}$$

then it’s easy to see how they can be put in one-to-one correspondence with the elements of $\mathbb{N}$:

Even though we’re burning through natural numbers at twice the rate of positive integers, we won’t run out. For any element of $\mathbb{Z}$, we can always find a corresponding element of $\mathbb{N}$.

We say that $\mathbb{Z}$ is countably infinite — which sounds like an oxymoron — as is any other set that can be put into one-to-one correspondence with $\mathbb{N}$.

Time to get real

Can we do something similar with the reals? (Spoiler: no.) It’s easy to get an intuitive sense of why they may not be countably infinite when we try to write down a description, as we did with $\mathbb{Z}$. In fact, let’s make things easier by just focusing on the interval $[0, 1]$:

$$D = \{ 0 \dots 0.0625 \dots 0.125 \dots 0.25 \dots 0.5 \dots 1 \}$$

Ugh, there are ellipses everywhere, like a plague of measles. The naturals were infinite at one end, and the integers were infinite at two ends (which we worked around by “folding them in half”), but the reals are infinite everywhere. Between any two real numbers there are an infinite number of real numbers. But maybe there’s a clever way to rearrange them so they can be counted? Imagine for a moment that there is. Imagine that we’ve constructed a list of the reals, $r_1, r_2, r_3$, etc. Never mind what order they’re in or what their values are.

Now imagine a number $r_0$ such that the first digit of $r_0$ is different from the first digit of $r_1$, and the second digit of $r_0$ is different from the second digit of $r_2$, and so on for every $r_n$. Then $r_0$ can’t be the same as any number in our original list because it differs in at least one place. Thus, $r_0$ is not in our original list, which is a contradiction to the notion that we had a list of every real number.

Thus, our interval $[0, 1]$ is uncountably infinite. It’s a bigger infinity than the integers. Getting back to the song, that means for interpretations $C$ and $D$, Sir Sly can’t count “all the numbers between zero and one.”

Where does that leave us? Where does that leave Sir Sly? For interpretations $A$ and $B$, “counting all the numbers between zero and one” is trivial. For $C$ and $D$, it’s literally impossible. Which interpretation you choose depends on how you interpret the rest of the song. And because the rest of the song is in English, there are even more tangled meanings to decipher. Is Sir Sly hopeful? Regretful? Inconsolable? Relieved?

A rational approach

There’s another option for “numbers,” the rationals, $\mathbb{Q}$, which we skipped over before. The rationals are so called because they are defined as a ratio between two integers, what we usually think of as fractions, and not because they’re more “sensible” than other numbers. (Q stands for quotient.) Let’s define two more sets, $F$ and $G$ ($E$, remember, was elephants) as

$$
F = \{ x |~x \in \mathbb{Q}, 0 < x < 1 \}, \\
G = \{ x |~x \in \mathbb{Q}, 0 \leq x \leq 1 \}$$

The rationals, like the integers, are countably infinite. Unlike with the integers, however, their unbounded and bounded subsets between zero and one are also countably infinite. They strike a middle ground between trivial and impossible. They are countable, but not in a finite amount of time. To count all the rational numbers between zero and one is to commit to a process.

If Sir Sly is referring to the rationals, then “counting all the numbers between zero and one” is an eternal promise, a dedication to a task that is neither trivial nor impossible, but all-consuming and forever, which sounds a lot like love.

Choosing one of these sets — $A$, $B$, $C$, $D$, $F$ or $G$ — brings clarity but also locks us into a single interpretation of the line from the song, which may not be what Sir Sly intends. Maybe the line is meant to be ambiguous, to change meaning from listen to listen, as it did for me, or even from one instance of the chorus to the next. Human beings are complex, human relationships even more so.

Is love rational? Is heartache real? Is forever too long to wait? What’s your interpretation?

Note: the notion of sizes of infinities and the first proof that the reals are uncountably infinite are due to German mathematician Georg Cantor, who also pioneered modern set theory. I first encountered them in George Gamow’s dated but still excellent One Two Three … Infinity, a recreational mathematics book about one notch up in sophistication from this article.

Tony Pisculli

Tony is fascinated by the intersection of language and mathematics. He has a BA in Mathematics from University of California at Santa Barbara and worked as a software engineer at Microsoft before decamping for the arts. Currently he is a writer, a fight director and the Artistic Director of the Hawaii Shakespeare Festival.

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