Hilbert's Paradox of the Grand Hotel: Another Look at Infinity
Hilbert’s Paradox of the Grand Hotel
When looking at mathematics, we can come across many startling concepts that are counterintuitive and appear to be false, but with careful thinking, they 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 in 1924. It is a thought experiment on the nature of infinite numbers, and it gives some surprising results.
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 in 1924. It is a thought experiment on the nature of infinite numbers, and it gives some surprising results.
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 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.
Moving the Guests
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 starts 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 still managed to find you a room!
Even though this infinite hotel was fully occupied, the manager still managed to find you a room!
No Such Thing as a Last 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.
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
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 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
Infinite Coaches of Infinite Guests
We can expand even further. This time, an infinite number of coaches arrive, each containing an infinite number of guests. How can all of these people fit into a fully occupied hotel?
Fundamental Theorem of Arithmetic
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 occupant's 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 2^n x 3^c. For example, the person currently in room 2 of the hotel will move to 2^2 x 3^0 = 4, and the person in seat 5 on bus 4 will move to 2^5 x 3^4 = 2592.
To see how this is useful in our example, we will number each coach, c, and number each coach occupant's 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 2^n x 3^c. For example, the person currently in room 2 of the hotel will move to 2^2 x 3^0 = 4, and the person in seat 5 on bus 4 will move to 2^5 x 3^4 = 2592.
Remaining Empty Rooms
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.
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 2^n x 3^c x 5^f. 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.
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 2^n x 3^c x 5^f. 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.
A Counterintuitive Challenge
The main reason why Hilbert’s hotel seems to present such a paradox is our understanding of finite versus infinite. Our everyday understanding of numbers is based on what we can see before us: the finite.
Take a finite list: for example, 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. In other words, 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.)
Take a finite list: for example, 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. In other words, 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.)
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