Hilbert's paradox of the Grand Hotel

Hilbert's paradox of the Grand Hotel is a mathematical paradox named after the German mathematician David Hilbert. Hilbert used it as an example to show how infinity does not act in the same way as regular numbers do.

The paradox

Normal hotels have a set number of rooms. This number is finite. Once every room has been assigned to a guest, any new guest that wants a room and does not have one yet cannot be served - in other words, the hotel is fully booked.

Now suppose that there is a hotel that has an infinite number of rooms. As a convenience, the rooms have numbers, the first room has the number 1, the second has number 2, and so on. If all the rooms are filled, it might appear that no more guests can be taken in, as in a hotel with a finite number of rooms. This is wrong, though. A room can be provided for another guest. This can be done by moving the guest in room 1 to room 2, the guest in room 2 to room 3, and so on. In the general case, the guest in room n will be moved to room n+1. After all guests have moved, room 1 is empty, and the new guest now has a room to occupy. This shows how we can find a room for a new guest even if the hotel is already full, something that could not happen in any hotel with a finite number of rooms.

In case of infinitely new guests

Another thing that can be done with this imaginary hotel is to double the number of people inside, again when all the rooms are already full. This is done by asking each guest to multiply their room number by two and move to that room. (If their previous room number was n, this time they would move to room number 2n.) This would send the guest in room 1 to room 2, the guest in room 2 to room 4, the guest in room 3 to room 6, the guest in room 4 to room 8, and so on. After finishing, we find that all the rooms with the odd numbers are empty. Then we can put an infinite number of guests into these empty rooms. Now the number of guests in the hotel has been doubled without making the hotel bigger.

If infinite groups of infinite guests come

The guest at room 11 would move to room 101. The second person of group 5 (Address 5-2) would go to room 25.

This is not really a paradox, it is only counterintuitive. In a normal hotel, with a finite number of rooms, the number of odd-numbered rooms, is smaller than the total number of rooms. In Hilbert's Hotel this does not seem to be the case.

In case of infinite vehicles of infinite groups of infinite guests

The guest 1 of group 2 of vehicle 1 (1-2-1) goes to room 121.

Further layers of infinity

Infinite aircraft carriers of the same infinite vehicles

Address 4-7-7-4 goes to room 4774.

Infinite spacecraft of the same infinite aircraft carriers.

Address 0-1 (hotel dweller) stays because 1-0-0-0-1 moves to room 10,001.

and so on.

Infinite layers of nesting

Each pod holds 10 people.

Each megapod holds 10 pods. (100 people)

Each supermegapod holds 10 megapods. (1,000 people)

Each superdupermegapod holds 10 supermegapods.

Each ultrasuperdupermegapod holds 10 superdupermegapod.

Each ultrasuperduperubermegapod holds 10 ultrasuperdupermegapods. (1,000,000 people)

And so on. This assumes that there is never an infinitieth layer. (The main ship)

Analysis

Hilbert's paradox is a veridical paradox: it leads to a counter-intuitive result that is provably true. The statements "there is a guest to every room" and "no more guests can be accommodated" are not equivalent when there are infinitely many rooms. An analogous situation is presented in Cantor's diagonal proof.

At first, this state of affairs might seem to be counter-intuitive. The properties of "infinite collections of things" are quite different from those of "finite collections of things". The paradox of Hilbert's Grand Hotel can be understood by using Cantor's theory of transfinite numbers. In an ordinary (finite) hotel with more than one room, the number of odd-numbered rooms is obviously smaller than the total number of rooms. However, in Hilbert's aptly named Grand Hotel, the quantity of odd-numbered rooms is not smaller than the total "number" of rooms. In mathematical terms, the cardinality of the subset containing the odd-numbered rooms is the same as the cardinality of the set of all rooms. Indeed, infinite sets are characterized as sets that have proper subsets of the same cardinality. For countable sets (sets with the same cardinality as the natural numbers) this cardinality is [math]\displaystyle{ \aleph_0 }[/math].

Put differently, for any countably infinite set, there exists a bijective function which maps the countably infinite set to the set of natural numbers, even if the countably infinite set contains the natural numbers. For example, the set of rational numbers—those numbers which can be written as a quotient of integers—contains the natural numbers as a subset, but is no bigger than the set of natural numbers since the rationals are countable: there is a bijection from the naturals to the rationals.

Other websites

• The Infinite Hotel Paradox - Jeff Dekofsky - TED-Ed Lessons