Question: How big is infinity?

Cathal Cummins answered on 18 Nov 2013:

As Michael says, there are lots of different infinities. One way to get a feeling of how counterintuitive the concept of infinity of the Natural numbers (1,2,3,4,…) is by the following example:

Imagine a hotel (Let’s call it the Hilbert Hotel) with an infinite number of rooms, each of which are occupied by a person. Using our intuition of _finite_ numbers, we might be led to believe that, since each room has a person in it, we would not be able to take on another guest… but we would be wrong! How is this possible — our hotel is fully-booked isn’t it?

Well this is where our ideas of counting break down. We are treating our infinity hotel as if it were a hotel with a _finite_ number of rooms. So let’s see how the Hilbert Hotel can take in new guests…

So say a new guest arrives and is looking for a room, the receptionist would look at the register and say

“I’m sorry all our rooms are taken… :(”

but then he would say,

“Oh wait, I can ask our guest in room 1 to move to room 2 and our guest in room 2 to move to room 3 and so on, which means that you can take room 1 …. :)”

And the guest will have a room in a fully booked hotel! Such a hotel simply couldn’t exist if it had a finite number of rooms.

We can extend the Hilbert Hotel to investigate all sorts of other weird things that happen when we have an infinite number of rooms — fascinating question, and one that has only been answered relatively recently as Michael says! 🙂