Imagine you’re the manager of a hotel with an infinite number of rooms. You may think that you can accommodate everyone who shows up but you are wrong. Hypothetically, there is a chance of you running out of space. How is that possible in an infinite hotel? Let’s find out.
To start with, let’s imagine that all the rooms are full with at least one person. Your hotel has an infinite number of people in infinite number of rooms. Now, a new person enters the hotel and wants a room. How do you accommodate him? Well, you just ask everyone to move down a room. The person in Room 1 moves to Room 2, the person in Room 2 moves to Room 3 and so on and so forth. When this happens, Room 1 gets vacated and you have an empty room for the extra guest.
The next day, if a bus having 100 people comes to the hotel, you’ll know exactly what to do. You just ask everyone to move down 100 rooms. So, the person in Room 1 moves to Room 101, the person in Room 2 moves to Room 102 and so on and so forth. 100 rooms get vacated for 100 more people and you can rest peacefully.
Your job gets complicated when a bus that’s infinitely long with an infinite number of people shows up the next day. Accommodating a finite number of people is easy, but accommodating infinite number of people is tricky. After some thought, you come up with a plan. You ask everyone to move to a room that’s double the number of their original room. So, the person in Room 1 moves to Room 2, the person in Room 2 moves to Room 4 and the person in Room 50 moves to Room 100. This will empty all the odd-numbered rooms and that will accommodate the people who have entered the hotel. For one more night, you can sleep with satisfaction.
The next day, you are in for a huge surprise. A party bus with an infinite number of people come in and they’re all named similarly. Every person is named with an infinite sequence of the letters A and B. So if one person is named ABABABABBBBAAAABABAA, the other is named BBAAABBAAAAAABA. Mathematically, accommodating all of them is impossible because you can’t have an algorithm that finds a room for everyone. The number of rooms in the hotel are countably infinite, but the number of people in the bus are uncountably infinite.
This post was inspired by a video on Veritasium.