Hilbert's Grand Hotel

Hilbert's Grand Hotel is an analogy and paradox used to explain the notion of countability. One starts off by imagining a hotel, with an infinite amount of rooms, and each is occupied. One's intuition says that it's not possible to fit any more people - however, due to the way infinite bijections work and the fact that they go against common sense, it is possible to still fit many more people.

Firstly, if there is a single new guest who wants a room, it is possible to accommodate them like so - namely, the hotel receptionist can tell everybody to move up one room, so the person checked into room zero moves to room one, the person checked into room one moves to room two, and so on. Because the set of rooms is never-ending, we don't run out of rooms and everybody who was checked in still has a room. Yet room number zero is now empty - the new guest can check in there. This is analogous to the proof that \(\omega\) and \(\omega+1\) are equinumerous.

Similarly, if someone in room \(n\) checks out of the hotel, then everybody in room \(m\) for \(m > n\) can move one room to the left, and all the rooms will be filled again.