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  • - 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)

- Imagine there's a hotel with infinite rooms.

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B1 US infinite room bus hotel infinite number letter

How An Infinite Hotel Ran Out Of Room

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    nao posted on 2021/06/23
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