Did you know that the entire number line is the same size as the piece of itself contained between zero and one?
Or that 1 + 1 – 1 + 1 – 1 + 1 – 1 + … ad infinitum is equal to half?
Welcome to the weird world of infinity – where nothing is as it seems, and no answer at all can sometimes be the best one you can get.
Going on holiday with David Hilbert
One of the most famous thought experiments to help understand the bizarreness of infinity was devised by David Hilbert, the legendary mathematician responsible for setting forth the 23 questions that would define 20th-century math. It’s become known as Hilbert’s Grand Hotel, and it usually goes something like this:
Imagine you’re on a trip, and you want to turn in for the night. You finally find a hotel – the only one for miles around – but alas, you see the sign says “no vacancies.” But look, you need somewhere to sleep, and the hotel looks pretty big, so you decide to go in and ask for a room anyway – just in case.
“We’re completely full,” says the check-in clerk, “not a single free room in the house.”
Disappointed, you turn to leave, but she stops you.
“Wait!” she says. “We can still fit you in – you see, this is a particularly special hotel. It has an infinite number of rooms. All we have to do is tell everybody already staying in the hotel to move into the next room over!”
She presses a button and speaks into the intercom.
“This is a customer announcement,” she says. “The guest staying in room one must please move into room two. The guest in room two is to move into room three. The guest in room three, please move to room four, and so on.”
She turns back to you with a smile on her face.
“There we go,” she says. “Room one should now be free. I’ll check you in.”
The Grand Hotel: no vacancies, but there’s room for everyone
So, you take your key and turn to leave for your room. But then, you see a gaggle of tourists come through the door.
“Hi,” the leader of the group says to the clerk. “There’s 20 of us – we heard this place can always make room for a few more.”
“That’s right,” replies the clerk. “Let me just shuffle some guests around.”
She goes to the intercom once again.
“Another customer announcement,” she says. “We have 20 new guests at the hotel. Can everybody please move to the room 20 to their right: room one please move to room 21, room two to room 22, and so on. Thank you!”
She turns back to the group.
“That should do it!” she says and checks the group into rooms one through 20.
The grand bus comes to the Grand Hotel
You have to admit, you’re impressed with this clerk: she’s managed to fit 21 new guests into a completely full hotel without breaking a sweat. But then, the phone rings, and you see her look worried.
“You’re sure? Yes, ma’am, we – we’ll see what we can do,” she says, and hangs up.
“There’s a full bus of travelers headed this way,” she tells you. “There’s infinitely many of them, and they all need a bed for the night – we’re going to need to double the number of rooms in the hotel to fit them all in!”
You both think for a while about how to solve this hospitality nightmare until suddenly it hits you.
“I’ve got it!” you tell the clerk. “Just send the guest in room one to room two, the guest in room two to room four, the guest in room three to room six, and so on. If every guest moves to the room whose number is twice their current room number, then there will be an infinite number of vacancies created, and everybody on the bus can have a room!”
“By jiminy, you’ve cracked it,” cries the clerk, firing up the telecom once again. “For that, the room is free.”
So what does Hilbert’s Hotel tell us, other than that hospitality workers are criminally undervalued as a profession? The big lesson, mathematically speaking, is that the thing we call “infinity” doesn’t behave as normal numbers do – and neither should it.
Infinity in math
Infinity is – well, it’s in the name: infinite. We’ve already seen how it’s unaffected by adding or multiplying constants, but what if we want to get really abstract about it? What if we wanted to find the sum of two infinities? Something like:
Well, this is actually fine, mathematically speaking. Think about it: you have something unfathomably large, and you add it to something else unfathomably large – what are you going to get?
Using the same reasoning, we can also multiply two infinities together.
But! What if we wanted to know the answer to this:
Well, that’s where things get complicated. Let’s take a look at why.
There are different types of infinity
Infinity is infinitely big, but some infinities are bigger than others. I know, that sounds completely bananas, but it’s true – and actually, you already knew it.
Hilbert’s Hotel showed us probably the simplest way to think about infinity: just plop yourself at one, and start walking up the number line. This is called countable infinity, and it’s the smallest type of infinity there is.
The term “countable infinity” may sound a bit like an oxymoron – how can something be countable and infinite, right? But the name isn’t meant to imply that you could ever count all of the members of this infinite set – it just refers to the idea that there’s some way that you could put them in a list. The most obvious countably infinite set is the natural numbers, which can be listed like this:
So, we’re already getting a sense here of something important when it comes to understanding infinities. Notice we’re talking about sets, and members of sets, and counting, rather than anything resembling a number. See, one of the biggest misconceptions when it comes to understanding infinity is to think of it as a really, really big number – the biggest number it’s possible to conceive of. But that’s not true.
There might be some of you out there saying something like “of course it’s not a number! That’s why we call it ‘infinity’ instead of, I don’t know, two or something!” But it’s easier to fall into this trap than you might think. After all, even in college-level math, we’re often encouraged to think of infinity as the limit of a sequence of ever-increasing numbers.
But infinity is not a number, and it doesn’t reliably behave like one – and that quickly becomes clear when we try to do math with it.
What does infinity minus infinity equal?
We’ve seen that the natural numbers – one, two, three, four, five, and so on – form a countably infinite set. And in fact, any infinite set with one-to-one correspondence to the natural numbers – that is to say, any infinite set where you can think of a sensible way to list the elements from one to … well, from one onwards – is the same: countably infinite. So, for instance, the set “even numbers” is countably infinite, because we can list them like this:
Or how about the set “integers” – you need a little bit of thought for this one.
Even “fractions” is countable, although the method for listing them is something rather unexpected:
And this is the logic that lies behind Hilbert’s Hotel: the set of “rooms in the hotel” is countable, since (as is traditional for hotel rooms) they are labeled by the natural numbers. When the guests all move one room higher, that’s the equivalent of labeling the rooms like this:
And when they move to the rooms twice their current room number, we can think of that as labeling them like this:
But here’s a question: what about the set of real numbers?
If you need a refresher, a real number is any number you would think of when somebody says “think of a number”. One, –72, ?, log(14) – if you can point at it on the number line, then it’s a real number. Generally, we write real numbers as decimal expansions, but does that help us list them?
Let’s make life easy for ourselves and only list the real numbers that are zero or above. The first number on the list is easy:
But what comes next? 1? 0.1? 0.000001? Something else?
The real numbers, it turns out, simply can’t be put in a list like the natural numbers or integers. They are, in mathematical terminology, uncountably infinite. And uncountable infinity is larger than countable infinity. Significantly larger, in fact.
So this gives us an answer to our earlier question – at least, part of an answer. We can take the difference of infinities, as long as we keep to some certain requirements. We can say, for instance:
But as for
Well, it can equal one:
Or negative pi:
There’s really no good answer at all – we’re stuck! (In math, we call such expressions “undefined”, which makes the whole situation much less embarrassing.)
Think Get confused like a mathematician
If you don’t have a headache yet from all this, then good news: we’ve basically just scratched the beginning of the surface of the weirdness that is infinity. But before we call it quits for now, let me ask you something: can you think of a set in between countable and uncountable infinity?
It’s a real question, by the way – nobody knows the answer. This problem – whether there exists any set that has a size larger than the natural numbers but less than the reals – is called the continuum hypothesis, and it has been sitting there unproven, taunting logicians, for nearly 150 years now. Unlike the Riemann hypothesis or P vs NP, this isn’t one of those “not proven but everybody basically thinks it’s true” hypotheses either – mathematicians are genuinely split on the question.
The problem, and the reason that we’re unlikely to see a solution any time soon, is that the continuum hypothesis is unprovable.
That’s not hyperbole: it is literally impossible, using the mathematical tools that we have at the moment, to prove the continuum hypothesis either way. Which sounds strange, right? How can it be possible to know for sure that a hypothesis can’t be proved – not that it’s just too difficult for anybody right now, but that even the brainiest person on Earth, given access to every piece of information known to humankind, would never be able to find a solution?
Well, it wasn’t easy, that’s for sure. Proving the unprovability took two world-renowned mathematicians more than three decades of wading through one of the most abstract and esoteric areas of math available. We’re not going to go into details here because, well, we don’t have 33 years free to explain it, but the CliffsNotes version goes like this:
In other words, we can’t prove it’s not true, we can’t prove it is true, let’s call the whole thing off.
Unsatisfying, I know – but there’s still a lesson to be learned here: that however confusing you might find the concept of infinity, at least know that you’re not alone. Because when it comes to infinity, for two of the world’s greatest mathematicians, “welp, guess we’ll never know!” was once a good – nay, a great – result.