The Infinite Hotel

Imagine a hotel with an infinite number of rooms. The rooms are simply numbered 1,2,3...ad infinitum. Each room will fit only 1 person in, and each room is full.

Now, what happens if one more person comes? Which room do you put them in?

I'll put him in room #1.
That's what I would have done.

And the person who was in room 1 before?

Originally posted by borissa
And the person who was in room 1 before?

And what if infinity more people come, each wanting a room?

This problem is called "Hilbert's Hotel" after the famous German mathematician David Hilbert. It's meant to illustrate the supposed paradox that an infinite set A can be in be in bijective correspondence with a proper subset B of A.

Of course, it's not really a paradox at all. You just have to think about it a bit.

(Here "bijective correspondence" means the elements of A can be paired off with the elements of B, with no element of B used twice and no elements of B left over. It's the correct notion for two sets A, B to "have the same number of elements".)

Edit: If you want an answer to the question put the original people in rooms 2, 4, 6, ... and the new people in rooms 1, 3, 5, ...

Originally posted by borissa
Now, what happens if one more person comes?

Shhhhhhh.... 😛

An interesting aspect of this 'paradox' is the following:

You have a hotel with infinite number of rooms and they're all filled up. You rent the rooms night after night.

A new guest comes. You place every guest in room #n in room #(n+1). That's the solution, right?

But the room #1 is not free before all guest move one room upwards. This takes infinit amount of time. So the new guest cannot ever get his room. Certainly not before the night is over, anyway.

So if you don't have infinite time at your disposal - the 'paradox' has not a solution. It simply can't be done.

It wouldn't take infinite time, if everyone moved simultaneously. There is a corridor between the rooms, so they can all move out of their rooms into the corridor, then all take 10 steps right, then all move into the next room. Say, 30 seconds?

Originally posted by borissa
It wouldn't take infinite time, if everyone moved simultaneously. There is a corridor between the rooms, so they can all move out of their rooms into the corridor, then all take 10 steps right, then all move into the next room. Say, 30 seconds?

Originally posted by SPMars
Edit: If you want an answer to the question put the original people in rooms 2, 4, 6, ... and the new people in rooms 1, 3, 5, ...

answer: if infinite people are coming to your hotel, and each person has to pay, you must be making a lot of money, so just build another hotel. 😛


I WIN!!!!!!

Originally posted by borissa
Imagine a hotel with an infinite number of rooms. The rooms are simply numbered 1,2,3...ad infinitum. Each room will fit only 1 person in, and each room is full.

Now, what happens if one more person comes? Which room do you put them in?

Originally posted by uzless
come on now, you can never fill an infinite amount of rooms. There will always be room. To say that each one is filled is an error in logic. Just cause you "say" the rooms are filled, it doesn't mean they are actually filled.

If I say this apple is a banana, it doesn't mean the apple is a banana. It just means my logic is flawed. To say that an infinite amount of rooms can be filled is also flawed.