ABSTRACT
Comment: It is clear that many other variations on this theme could be devised, and such variations have long been familiar, as we shall see. This one derives from a suggestion made by Denyer.5
(iv) The paradox of the hotel Suppose there is a hotel with infinitely many rooms, each occupied at a particular time. Then a newcomer can be accommodated without anybody having to move out; for if the person in the first room moves into the second, and the person in the second room moves into the third, and so on ad infinitum, this will release the first room for the newcomer. Indeed infinitely many newcomers can be accommodated without anybody having to move out; for if the person in the first room moves into the second, and the person in the second room moves into the fourth, and the person in the third room moves into the sixth, and so on ad infinitum, this will release the infinitely many odd-numbered rooms. And if, when all the guests have settled into their new rooms, each is dismayed by how small a bar of soap has been left in the washbasin, then they can systematically shunt bars of soap along the rooms to ensure that each has two bars instead, or indeed a hundred. All of this puts, to say the least, a strain on our intuitions.
