# Hilbert's Hotel

Hilbert's Hotel is a famous story about infinity attributed to David Hilbert (1862-1943).

Is it documented that Hilbert's Hotel is in fact due to Hilbert, and if yes, where?

• arxiv.org/abs/1403.0059 Sep 19 '14 at 8:55
• This could have been asked on Math.SE Sep 21 '14 at 0:03

How is this for an infinite quantity? Let us take as the simplest example the quantity of integer numbers. Here already this law "a part is smaller than the whole" no longer holds. We can explain this important fact easily using our example of the occupied hotel. This time we assume that the hotel has infinitely many numbered rooms, $1,2,3,4,5,\ldots$, in each of which there lives a guest. When a new guest arrives, all the manager has to do is to allow each of the old guests to occupy the room having one number higher, and this will free room number 1 for the new arrival. Of course, in this way space can be made for any finite number of new guests, and in this world of an infinite number of houses and occupants there will be no housing shortage. [...]
Indeed, it is even possible to make space for an infinite number of new guests. For example, each of the old guests, originally occupying room number $n$, just has to move to number $2n$. Then the infinite number of rooms with odd numbers become free for new guests.