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Hilbert's Infinite Hotel

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David Hilbert (1862-1943), a German mathematician, was interested in geometrical spaces and number theory. He proposed the Infinite Hotel, otherwise known as Hilbert's Hotel, as an illustration of the problems of treating infinity as a number.

The hotel has an infinite number of rooms and is always full, but despite this can cater for any number of extra guests that arrive.

The Hotel

Imagine that there is a hotel with an infinite number of rooms, all laid out in a row (an unlikely scenario, but press on). Now imagine that we fill every one of the rooms with one person each, so the hotel is full. To reinforce this, we light up a flashing neon 'Hotel Full' sign out in front. But because this is Hilbert's Infinite Hotel, we also put a 'Rooms Available' sign out the front beside the other one.

So, is the sign right? 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.

A Single Extra Guest

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. The person in room 1 moves to room 2; the person in room 2 moves to room 3; 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. This leaves room 1 free so the newly-arrived person will take room 1.

A Bus-load of Extra Guests

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 (ie 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...

An Infinite Bus

Q: A bus containing an infinite number of guests arrives. The guests pile out and crowd into reception, demanding rooms. Is it possible to keep them happy?

A: Yes, it is possible: get everybody to move to the room which is double the room number they have already (ie 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 accommodate an infinite amount of new guests. With this case, we show that two times infinity (or, in fact, any finite number times infinity) is still infinity.

But wait...

An Infinite Convoy of Buses

Q: An infinite number of buses, each containing an infinite number of guests, arrives. The guests all asks for rooms. Can space be found for them?

A: Yes, it is possible, but it's a little complicated. First we do the trick we did in the previous section - moving every guest who already has a room into the room with twice the number. This gives us an infinite number of empty rooms in the hotel.

We number each of the buses 1, 2, 3 and so on. We also number the people on each bus 1, 2, 3 and so on.

  • The 1st person on the 1st bus goes into the first empty room.

  • The 2nd person on the 1st bus and the 1st person on the second bus get the next two empty rooms.

  • The 3rd person on the 1st bus, the 2nd person on the 2nd bus and the 1st person on the 3rd bus get the next three empty rooms.

  • The 4th person on the 1st bus, the 3rd person on the 2nd bus, the 2nd person on the 3rd bus and the 1st person on the 4th bus get the next four empty rooms.

  • Let's hope you can see how this is going...

With this system, everybody gets a room and the hotel is still full. This shows that infinity multiplied by infinity is still infinity.

Bigger Infinities

It is clear from all this that no matter what you do to infinity, you always end up with infinity. Is there such a thing as an infinity which is bigger than the infinity we're talking about here?

This question was posed and answered by Georg Cantor, who showed that such an infinity does exist. He used his ingenious 'Diagonal Argument' to prove the existence of bigger and bigger infinities. But that's another story.

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