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Hilbert's Paradox of the Grand Hotel: Another Look at Infinity

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I am a former maths teacher and owner of Doingmaths. I love writing about maths, its applications and fun mathematical facts.

When looking at mathematics, we can come across many startling concepts that are counterintuitive and appear to be false, but with careful thinking can be proven to be mathematically true. We call these veridical paradoxes, examples of which include the Monty Hall problem and Schrödinger’s cat. Hilbert’s Paradox of the Grand Hotel is another such example.

Also known as the ‘Infinite Hotel Paradox’ or ‘Hilbert’s Hotel’, the Paradox of the Grand Hotel was first introduced by the German mathematician David Hilbert (1862–1943) in a lecture of 1924. It is a thought experiment on the nature of infinite numbers which gives some surprising results.

David Hilbert (1862–1943)

David Hilbert (1862–1943)

Arriving at the Infinite Hotel

Suppose after a long day on the road, you arrive at the Grand Hotel exhausted and in dire need of a shower. The hotel has a large sign out the front boasting of its infinite number of rooms, but unfortunately all of the rooms are occupied. You are about to leave when the manager tells you that this isn’t a problem; he can find room for you.

He asks the guest in room number 1 to move into room number 2. He asks the guest from room 2 to move into room 3 and so on. If a guest started in room n, they move into room n+1. He then hands you the key to room 1. Even though this infinite hotel was fully occupied, the manager has still managed to find you a room.

We can continue this idea further. If five guests arrived simultaneously, the manager could move the guest from room 1 into room 6, room 2 into room 7 and so on, each guest moving five rooms up, which would leave five spare rooms for the new guests.

Note how the manager can't just give new guests the last room/rooms. As there are an infinite number of rooms, there is no such thing as the last room; you can always count higher.

Everybody Moving Up a Room

Everybody Moving Up a Room

An Infinite Number of New Guests

But what about if an infinite number of guests appeared looking for rooms? This isn’t a problem either. This time the manager would simply ask each current room occupant to move to the room that is double theirs, so room 1 moves to room 2, room 2 moves to room 4 and so on, each guest moving from n to 2n. This would leave the odd-numbered rooms free. As there are an infinite number of odd numbers, our infinite number of new guests can then move into these.

Everybody Moving to 2n

Everybody Moving to 2n

An Infinite Number of Coachloads of Guests

We can expand even further. This time, an infinite number of coaches arrive, each containing an infinite number of guests. How can we fit all of these people into our fully occupied hotel?

There are several ways of doing this, but we are going to look at a way that utilises prime factorisation and a very useful theory known as the Fundamental Theorem of Arithmetic. This states that for every whole positive number larger than one, we can write the number as the product of its prime factors and that this product is unique (ignoring rearranging the same numbers into different orders). For example, 72 = 2 x 2 x 2 x 3 x 3. There is no way of writing 72 as a product of different prime numbers and obviously the product of 2, 2, 2, 3 and 3 will always equal 72.

To see how this is useful in our example, we will number each coach, c, and number each coach occupants seat number n. We will also represent current guests of the hotel as being on coach 0. Each person now moves into the room given by the product 2n x 3c. E.g. the person currently in room 2 of the hotel will move to 22 x 30 = 4, the person in seat 5 on bus 4 will move to 25 x 34 = 2592.

By the fundamental theorem of arithmetic mentioned above, each person must move to their own room. There will be nobody without a room and no room will be double booked. Hence the hotel can accommodate everybody. Note that there will also be a lot of empty rooms, for example the number 15 cannot be written as a product of twos and threes. For the scope of this article however, the important thing is that all of the guests have been given their own room.

Further Levels of Infinity

The amazing thing about Hilbert’s hotel is that we can continue with further examples. Suppose now that the hotel is based on the bank of a river and across the river come an infinite number of ferries, each carrying an infinite number of coaches with an infinite number of guests on each. To house these guests in the hotel we can expand our previous method and use the next prime number, 5. So now the guest sitting in seat n, on coach c, on ferry f will go into the room numbered 2n x 3c x 5f. Again, each guest will get their own room in our fully occupied hotel.

We can continue this pattern for further infinites, such as an infinite number of rivers each containing an infinite number of ferries and so on by adding further prime numbers to our product. One interesting side point however is that although we can continue to add further layers of infinity, we cannot have an infinite number of layers of infinity. Our hotel will not be able to accommodate this.

Why Does Hilbert's Paradox Seem So Counterintuitive?

The main reason why Hilbert’s hotel seems to present such a paradox is our understanding of finite versus infinity. Our everyday understanding of number is based on what we can see before us; the finite. For a finite list, e.g. the whole numbers from 1 to 100, if we take a subset of this such as the odd numbers, then that subset is smaller than the original set i.e. there are fewer odd numbers between 1 and 100 than there are whole numbers.

With infinity this is no longer the case. The size of the set of whole positive numbers is exactly the same as the size of the set of whole positive odd numbers. For further reading on this, visit my article on the different sizes of infinity.

Sources and Further Reading

This content is accurate and true to the best of the author’s knowledge and is not meant to substitute for formal and individualized advice from a qualified professional.

© 2021 David

Comments

David (author) from West Midlands, England on February 09, 2021:

Imagine the plumbing :-)

Tanmoy Acharya 100 on February 09, 2021:

The really interesting thing about this grand hotel, would be this: If we could somehow manipulate the number of toilets in the hotel.