There is a hotel with an infinite number of rooms. They are numbered one, two, three, four and so on "ad infinitum". This is the Hilbert Hotel and you are the manager. Now you might think you could accommodate everyone who ever shows up, but there is a limit, a way to transcend even the infinity of rooms in the Hilbert Hotel.
Let us say only one person is allowed in each room and all the rooms are full. There is an infinite number of people in an infinite number of rooms. Then someone new shows up and wants a room, but all the rooms are full. So what are you supposed to do? Well, a bad manager might turn them away, but you know about infinity. So you tell all the guests to move one room over.
So the person in room one moves to room two. The person 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 with a hundred people turns up, you know exactly what to do: you move everyone down a hundred rooms and put the new guests in the rooms that have become vacant. But let us assume a bus arrives that is infinitely long and carries an infinite number of people.
You know what to do with a finite number of people, but what do you do with an infinite number of people? You think about it for a minute and then you have a plan. You instruct each of your existing guests to move to the room with the duplicate room number. So the person in room one moves to room two, room two moves to room four, room three moves to room six, and so on. And now all the odd-numbered rooms are free. And you know that there are an infinite number of odd numbers. So you can give each person on the infinite bus a unique room with an odd number.
This hotel really starts to make you feel like it could basicallly fit almost anyone and that's the beauty of infinity, it goes on and on. And then suddenly more infinite buses appear, not just one or two, but an infinite number of infinite buses. So what can you do? Well, you use an infinite spreadsheet, of course. You make a row for each bus, bus 1, bus 2, bus 3, and so on. And at the top is a row for all the people who are already in the hotel. The columns represent the position that each person occupies. So you have hotel room one, hotel room two, hotel room three, and so on. And then bus one position one, bus one position two, bus one position 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? Start in the top left corner and draw a line that zigzags across the table, going over each unique ID exactly once. Then imagine dragging the opposite ends of this line to straighten it. So we've gone from an infinite by-infinite grid to a single infinite line. It is then quite easy to assign each person in this row to a specific room in the hotel. So everyone fits in, no problem.
But now a large bus pulls up. An endless party bus with no seats. Instead, everyone on board is identified by their unique name, which is kind of weird. So their names are all just two letters, A and B. But each name is infinitely long. So someone is called A, B, C, D, E, and so on. Someone else is called AB, AC, AD, AE, AF.. and so on. On this bus, there is a person with every possible infinite sequence of these two letters.
Well, AB, BA, , Let's call him Abba for short. He comes to the hotel to set up the rooms, but you tell him, "I'm sorry, we can't accommodate all of you in the hotel." And he says: "What do you mean? "There are an infinite number of us, "and you have an infinite number of rooms. "Why doesn't that work?" So you show him. You get 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 AB AB AB AB AB AB and repeat. And so you continue, putting a different row of As and Bs next to each room number. "Now comes the problem," you say to ABBA, "let's say we have a complete, infinite list. "I can still write down the name of a person "who doesn't have a room yet." 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 change it from a B to an A. And you do this throughout the list. And the name you write down is guaranteed not to appear anywhere on the 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 differ from every name on the list by at least one letter. The letter is on the diagonal. The number of rooms in the Hilbert
Hotel is infinite, sure, but countably infinite. That is, there are as many rooms as there are positive integers from one to infinity. In contrast, the number of people on the bus is not countably infinite. If you try to assign an integer to each person, there will still be people left over. Some infinities are larger than others. So there is a limit to the number of people that can fit in the Hilbert Hotel. That's amazing enough, but what's even crazier is that the discovery of different-sized infinities sparked a series of investigations that led directly to the invention of the device you're reading this on.
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