# Infinity, and the Infinite Hotel Paradox

Created | Updated Feb 28, 2002

David Hilbert (1862-1943), a German mathematician, was interested in geometrical spaces and number theory. He proposed the Infinite Hotel Paradox, otherwise known as Hilbert's Paradox or Hilbert's Hotel. This paradox states: Imagine that there is a hotel with infinite rooms in a row (an unlikely scenario, but press on). Now imagine that we fill every one of rooms with one person each, so the hotel is full. To reinforce this, we light up a No Vacancies sign out in front.

So, is it possible to squeeze another person in?

To answer this, we look at the meaning of infinity. Infinity is not a number, despite what people may tell you; it's a concept. As we'll see, you can do a lot of squeezing of infinity without it moving much.

Q: A guest arrives at reception, and asks for a room. Is it possible to allocate him a room?

A: Yes, it is possible: get everybody to shift to the next room (i.e. the person in room 1 moves into room 2; the person in room n moves into room n+1). Since the hotel is infinite, there will always be a next-door to move into. The newly-arrived person will take room 1.

Q: Twenty guests arrive at reception, and they ask for rooms. Is it possible to allocate them rooms?

A: Yes, it is possible: get everybody to shift to the room twenty positions down (i.e. the person in room 1 moves to room 21; the person in room n moves into room n+20). The newly-arrived group will take rooms 1 to 20.

Now, you're thinking, this is a bit dodgy, isn't it? The hotel was full already! Well, this shows how infinity can't be regarded as a number in the normal sense, because infinity plus a finite number is still infinity.

But wait, there's more...

Q: A bus containing an infinite amount of guests arrive at reception, and they ask for rooms. Is it possible to allocate them rooms?

A: Yes, it is possible: get everybody to move to the room which is double the room number they have already (i.e. the person in room 4 moves to room 8; the person in room n moves to room 2n). The first person in the newly-arrived group gets room 1; the nth person gets room 2n-1.

This is especially weird. Even in a full hotel, you can accomodate an infinite amount of guests. With this case, we show that two times infinity (or, in fact, any finite number times infinity) is still infinity.

But wait...

Q: An infinite number of buses, each containing an infinite amount of guests arrive at reception, and they ask for rooms. Is it possible to allocate them rooms?

Stay tuned!

So, is it possible to squeeze another person in?

To answer this, we look at the meaning of infinity. Infinity is not a number, despite what people may tell you; it's a concept. As we'll see, you can do a lot of squeezing of infinity without it moving much.

Q: A guest arrives at reception, and asks for a room. Is it possible to allocate him a room?

A: Yes, it is possible: get everybody to shift to the next room (i.e. the person in room 1 moves into room 2; the person in room n moves into room n+1). Since the hotel is infinite, there will always be a next-door to move into. The newly-arrived person will take room 1.

Q: Twenty guests arrive at reception, and they ask for rooms. Is it possible to allocate them rooms?

A: Yes, it is possible: get everybody to shift to the room twenty positions down (i.e. the person in room 1 moves to room 21; the person in room n moves into room n+20). The newly-arrived group will take rooms 1 to 20.

Now, you're thinking, this is a bit dodgy, isn't it? The hotel was full already! Well, this shows how infinity can't be regarded as a number in the normal sense, because infinity plus a finite number is still infinity.

But wait, there's more...

Q: A bus containing an infinite amount of guests arrive at reception, and they ask for rooms. Is it possible to allocate them rooms?

A: Yes, it is possible: get everybody to move to the room which is double the room number they have already (i.e. the person in room 4 moves to room 8; the person in room n moves to room 2n). The first person in the newly-arrived group gets room 1; the nth person gets room 2n-1.

This is especially weird. Even in a full hotel, you can accomodate an infinite amount of guests. With this case, we show that two times infinity (or, in fact, any finite number times infinity) is still infinity.

But wait...

Q: An infinite number of buses, each containing an infinite amount of guests arrive at reception, and they ask for rooms. Is it possible to allocate them rooms?

Stay tuned!