Infinity

Generally speaking, people are not indifferent to math. When I introduce myself as a mathematician, I rarely get the disinterested “oh”s I’d expect if I were instead middle management at an unfamiliar, generically-named firm. Sometimes I get math horror stories, like when people hit their wall in algebra or calculus. (When I’m lucky, I get to hear about the special teacher who got them over it.) But just as often, I find that people are curious, that they have some lingering question they’ve always wanted to get down to the bottom of. Most often, those questions deal with infinity, like  “What’s $\infty$ divided by $\infty$?” As much as I love talking about math, these conversations are difficult for me — infinity is far too intricate a concept to elucidate through chit-chat. But I suppose I can try to do so here.

If I were to ask you what infinity times $1$ is, you’d tell me infinity (any number times $1$ is the same number, after all). If I asked for infinity times $2$, you’d probably still say infinity, because doubling a number makes it bigger, and infinity is already as big as it gets. Then we have two equations:

$$ \begin{align*} \infty\cdot 1 &= \infty \\ \infty\cdot 2 &= \infty \end{align*} $$

Let’s divide both of these equations through by $\infty$:

$$ \begin{align*} 1 &= \dfrac{\infty}{\infty} \\ 2 &= \dfrac{\infty}{\infty} \end{align*} $$

Here’s the problem: is $\dfrac{\infty}{\infty}$ equal to $1$ or $2$ (or something else)? There are lots of reasonable options for what $\dfrac{\infty}{\infty}$ could be, each as valid as the next. We can’t possibly decide between them all. Instead, we just have to agree that $\dfrac{\infty}{\infty}$ doesn’t really mean anything. In mathematics, we say that it is undefined.

If it seems odd that we just don’t assign a value to $\dfrac{\infty}{\infty}$, keep in mind that $\infty$ is not a number in our usual sense. The number system we deal in most often is called the real numbers. Real numbers obey all the rules of addition, subtraction, multiplication, and division that we’re used to. But $\infty$ is not a real number, so there is no expectation that we can use it like one. Mathematicians do have other number systems, though, and some include infinity. It’s like how, in the English language, cromulent is a word according to this dictionary but not this one. In mathematics, as in Scrabble, you need to establish the rules of the game up front.

One number system that includes infinity is the the ordinal numbers. But weird stuff starts happening in the ordinal numbers, like different ordinals having the same size. In the real numbers, if $x\neq y$, then one of $x$ and $y$ is bigger than the other. However, we can find different infinite ordinals that have exactly the same size. For example, consider the infinity that represents the number of natural numbers ($0$, $1$, $2$, $3$, etc.) and the infinity that represents the odd natural numbers ($1$, $3$, $5$, $7$, etc.). At first blush, it may seem that the first is larger because it includes all the even numbers as well, but they actually have the same size. To see why, we can list the elements of each and pair them up.

Since the two sets pair up perfectly, they have the same size.

In a world where two different infinities can have the same size, it can be quite difficult to show that two infinities have different sizes. In the late 19th century, Georg Cantor triumphed in doing just that — he showed that the infinity that counts all real numbers is larger than the infinity counting all whole numbers. He was even so bold as to suggest that the real-number-infinity was the next largest one after the whole-number-infinity, that there were none in between. It took mathematicians almost 100 years before they could resolve this claim, the so-called Continuum Hypothesis, but the answer was baffling: we can’t say. Not that we don’t know, but that we do know that we cannot determine whether it is true or not. It is possible, based on the rules of ordinal numbers that we’ve adopted, that there is no intermediate infinity. But those same rules also allow there to be all kinds of infinities in between. In a number system other than the ordinals with more stringent rules, maybe we’d be able to decide for sure whether intermediate infinities existed, but in this system, the question is undecidable.

The undecidability of the Continuum Hypothesis puts mathematicians on strange footing. On the one hand, we are happy to have finally resolved a really tough question. We had to develop some really sophisticated and powerful techniques to do it, and the man who finished the job, Paul Cohen, won a Fields Medal (think mathematical Nobel) for his efforts. But on the other, it’s hard to feel satisfied with so much ambiguity still floating around. And for those mathematicians who felt torn over this result, it kept piling on. For the next few decades, mathematicians kept showing that some of the most exciting and compelling conjectures in the field were undecidable. It started to feel like whole areas of math were turning up devoid of fulfilling results.

And then recently, there was a breakthrough: Maryanthe Malliaris and Saharon Shelah announced that $\mathbf{p}=\mathbf{t}$. These quantities, $\mathbf{p}$ and $\mathbf{t}$, are infinities between whole-number-infinity and real-number-infinity. I won’t go into detail about exactly what they represent because the precise statements are hard to wrap your head around without considerable background. (I am taking a risk here: John Baez recently lampooned a Quanta article about this discovery with the headline “MATHEMATICIANS MAKE IMPORTANT DISCOVERY: But Details Don’t Matter”. Contra Baez, I think that the context is more relevant than the content to the non-specialist here.) Suffice it to say that $\mathbf{p}$ and $\mathbf{t}$ live squarely in the murky gap of the Continuum Hypothesis, where it has seemed for so long that anything could be happening. Jaded by so many years of undecidability proofs, lots of mathematicians just expected that eventually we’d see the same thing play out here. (Tim Gowers, a former Fields Medalist himself, seems to suggest that a defeatist attitude may have held back progress on the problem until now.) But Malliaris and Shelah persisted and defied expectations and showed that we can still find satisfying new truths about these infinities.

You see, mathematicians hit their walls, too. But we’re beaming right now because we’ve just gotten our first glimpse of the other side.