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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

• 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.

• 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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# How An Infinite Hotel Ran Out Of Room

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nao posted on 2021/06/23
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