Hilbert's Hotel

Operations at the hotel

Suppose the hotel has infinitely many rooms numbered with the natural numbers, room 1, room 2, room 3 and so on, and that every room has a guest.

• When one guest arrives: the manager asks everyone to "move to the room whose number is your current room number plus 1". The guest in room n moves to room n+1, and the new guest takes the vacated room 1. Not a single guest is turned out
• When infinitely many guests (a first, a second, a third and so on) arrive: everyone is asked to "move to the room whose number is twice your current room number". The existing guests fit into the even-numbered rooms, the infinitely many odd-numbered rooms become vacant, and the new guests are put into them in order
• When infinitely many buses each carrying infinitely many new guests arrive: everyone can still be accommodated, for example by an assignment using prime numbers (the existing guest in room n goes to room 2ⁿ, the nth passenger on the first bus to room 3ⁿ, on the second bus to room 5ⁿ, and so on, each group assigned to the rooms that are powers of a different prime). The fact that a great many rooms are left empty is also part of why this is often used as an example of the strangeness of infinity

Mathematical meaning

What is happening here is the fact that a one-to-one correspondence can be set up between an infinite set and a part of itself. The whole of the natural numbers and "the natural numbers from 2 up", or the whole of the natural numbers and the whole of the even numbers, can each be matched exactly, with nothing left over on either side. In Georg Cantor's set theory, sets that can be put in one-to-one correspondence are regarded as having "the same cardinality", so these are all infinities of the same size (countable infinity). Richard Dedekind, conversely, adopted this property (that a set can be put in one-to-one correspondence with a proper subset of itself) as the definition of an infinite set.

On the other hand, there are groups of guests that even this hotel cannot accommodate. Guests from an "uncountable" set that cannot be put in one-to-one correspondence with the natural numbers, such as the whole of the real numbers, cannot all be assigned room numbers however they are rearranged. This is the result shown by Cantor's diagonal argument, and it shows that infinities too differ in size. The parable of the hotel is often used in teaching as an entry point for explaining the properties of the concept of countable infinity.

How it came about

The source of this story was unclear for a long time. Some accounts give Hilbert's 1925 lecture "On the Infinite (Über das Unendliche)" as the source, but the text of that lecture, published in 1926, does not contain the hotel. According to the research of the historian of science Helge Kragh, Hilbert brought up the hotel in a lecture course at Göttingen in January 1924, but did not publish it as a paper. The later publication of the lecture notes made it possible to confirm that Hilbert himself had told it.

It became generally known after Gamow introduced it in his 1947 book "One Two Three... Infinity" (Japanese title "1, 2, 3…無限大"). Gamow cites as his source an unpublished "collection of Hilbert anecdotes" by Richard Courant. Gamow stayed in Göttingen in the late 1920s, and it has been suggested that he may have heard the story there.

Use in philosophy and cosmology

From the 1970s on, this thought experiment came to be used outside mathematics as well. In particular, those who argue that the past of the universe cannot extend infinitely have cited it as grounds for denying the existence of an actual infinite, in the form "if we assume that infinitely many things actually exist, absurdities like this hotel arise". Against this there is the objection that being counterintuitive and being logically impossible are different things.