Hilbert's Hotel

Infinity in Mathematics takes an altogether different meaning. Read my blog on Infinity I wrote earlier. The Hilbert hotel problem is an intuitive way to understand this paradox of infinite arithmetic. It was posed by German Mathematician David Hilbert. It goes like this

There is a hotel in which there are infinite number of rooms and there are infinite number of guests staying there. Now if one guest shows up then how will the owner accommodate this person ? Well since there are infinite number of rooms the owner can ask each of the guest to move to the next room. So the guest in room 1 will move to room 2, guest in room 2 to room 3 and so on and so the first room will be empty and he can put the guest in that room.

The second question he posed if there are countable infinite number of people show up then how will the owner make room for all these people. In this case the owner can ask each person to move to the room # twice their room #. So the guest in room #1 move to room #2, Guest in room #2 move to room #4 and guest in room #3 to room #6 and so on. So in a way we will have all odd number rooms be available and since there are infinite number of odd numbers we can move all our infinite number of guests to these rooms.

The third question he posed was what would happen if there are countable infinite number of groups, each containing countable infinite number of people. Then how can the owner accommodate these people. To understand this lets start with the basic fact that there are infinite number of prime numbers and power of each prime gives a number unique from the other power of prime. So we will first move everybody in the hotel into the rooms with power of 2, because they are infinite many and we have infinite many occupants. The next group will be moved into the room # starting with power of 3 ie room # 3, 9 ,27, 81.. and so on. Thus all the infinite group of people each containing infinitely many people will be placed into the hotel !