How An Infinite Hotel Ran Out Of Room

• Imagine there's a hotel with infinite rooms. They're numbered one, two, three, four, and so on forever. This is the Hilbert Hotel, and you are the manager. Now it might seem like you could accommodate anyone who ever shows up, but there is a limit, a way to exceed even the infinity of rooms at the Hilbert Hotel.

To start, let's say only one person is allowed in each room and all the rooms are full. There are an infinite number of people, in an infinite number of rooms. Then someone new shows up and they want a room, but all the rooms are occupied. So what should you do? Well, a lesser manager might turn them away, but you know about infinity. So you get on the PA and you tell all the guests to move down a room.

So the person in room one moves to room two. The one in room two moves to room three, and so on down the line. And now you can put the new guest in room one. If a bus shows up with a hundred people, you know exactly what to do: just move everyone down a hundred rooms and put the new guests in their vacated rooms.

But now let's say a bus shows up that is infinitely long, and it's carrying infinitely many people. You knew what to do with a finite number of people, but what do you do with infinite people? You think about it for a minute and then come up with a plan. You tell each of your existing guests to move to the room with double their room number.

So the person in room one moves to room two, room two moves to room four, room three to room six, and so on. And now all of the odd numbered rooms are available. And you know there are an infinite number of odd numbers. So you can give each person on the infinite bus a unique, odd numbered room. This hotel is really starting to feel like it can fit everybody.

And that's the beauty of infinity, it goes on forever. And then all of a sudden, more infinite buses show up, not just one or two, but an infinite number of infinite buses. So, what can you do? Well, you pull out an infinite spreadsheet of course. You make a row for each bus: bus 1, bus 2, bus 3, and so on.

And a row at the top for all the people who are already in the hotel. The columns are for the position each person occupies. So you've got hotel room one, hotel room two, hotel room three, et cetera. And then bus one seat one, bus one seat two, bus one seat three, and so on. So each person gets a unique identifier, which is a combination of their vehicle and their position in it.

So how do you assign the rooms? Well, start in the top left corner and draw a line that zigzags back and forth across the spreadsheet, going over each unique ID exactly once. Then imagine you pull on the opposite ends of this line, straightening it out. So we've gone from an infinite by infinite grid to a single infinite line.

It's then pretty simple just to line up each person on that line with a unique room in the hotel. So everyone fits, no problem. But now a big bus pulls up. An infinite party bus with no seats. Instead, everyone on board is identified by their unique name, which is kind of strange.

So their names all consist of only two letters, A and B. But each name is infinitely long. So someone is named A, B, B, A, A, A, A, A, A, A, A, A, and so on forever. Someone else is named AB, AB, AB, AB, AB, et cetera. On this bus, there's a person with every possible infinite sequence of these two letters.

Now, ABB, A, A, A, A, I'll call him Abba for short. He comes into the hotel to arrange the rooms, but you tell him, "Sorry, there's no way we can fit all of you in the hotel." And he's like, "What do you mean? There's an infinite number of us and you have an infinite number of rooms. Why won't this work?"

So you show him. You pull out your infinite spreadsheet again and start assigning rooms to people on the bus. So you have room one, assign it to ABBA, and then room two to AB AB AB AB repeating. And you keep going, putting a different string of As and Bs beside each room number.

"Now here's the problem," you tell ABBA, "let's say we have a complete infinite list. I can still write down the name of a person, who doesn't yet have a room." The way you do it is you take the first letter of the first name and flip it from an A to a B. Then take the second letter of the second name and flip it from a B to an A. And you keep doing this all the way down the list.

And the name you write down is guaranteed to appear nowhere on that list. Because it won't match the first letter of the first name, or the second letter of the second name, or the third letter of the third name. It will be different from every name on the list, by at least one character: the letter on the diagonal.

The number of rooms in the Hilbert Hotel is infinite, sure, but it is countably infinite. Meaning there are as many rooms as there are positive integers, one to infinity. By contrast, the number of people on the bus is uncountably infinite. If you try to match up each one with an integer, you will still have people leftover.

Some infinities are bigger than others. So there's a limit to the people that you can fit in the Hilbert Hotel. This is mind-blowing enough, but what's even crazier is that the discovery of different sized infinities sparked a line of inquiry that led directly to the invention of the device you're watching this on right now. But that's a story for another time.

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