– 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. (upbeat music).