There are two ways you can define the hotel as being full: 1) The hotel is “full” if all the rooms are occupied. 2) The hotel is “full” if there is no way to

There are two ways you can define the hotel as being full: 1) The hotel is “full” if all the rooms are occupied. 2) The hotel is “full” if there is no way to rearrange the existing guests to leave an empty room. These two definitions of “full” are equivalent for finite hotels. The ‘paradox’ is illustrating that these two definitions are not the same for infinite hotels. It serves as a useful reminder that there might be more “space” in infinite sets than we are used to in the finite case, and that if we are dealing with infinite sets we should keep in mind the fact that (1) and (2) (or whatever analogues are relevant for our problem) might be different, despite seeming the same based on our real-world experience.