[ajanuary]

It brings up an interesting point, though. If I have an infinite number of rooms and an infinite number of guests, then one might intuit that every room is occupied.

There is a mapping of every guest to a room: guest n is in room n. Yet somehow there exists a room that has no guest, despite being able to model both rooms and guests with the same infinite set.

[nhaehnle]

There is no empty room initially. However, by having all guests move simultaneously, you can make one room free without having any guest leave the hotel.

Incidentally, this is one way to define infinite sets. A set is infinite if and only if there exists a proper subset that has the same size (cardinality).