Is Infinity Real?

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Suppose a new guest arrives and wishes to be accommodated in the hotel. Because the hotel has infinitely many rooms, we can move the guest occupying room 1 to room 2, the guest occupying room 2 to room 3 and so on, and fit the newcomer into room 1. By repeating this procedure, it is possible to make room for any finite number of new guests." This ignores a significant point; in this case, the process of switching guests to new rooms is infinite, so although the new guest is settled in room 1, the infinite hallway between the rooms always contains a person switching rooms with the next guest in the line. This means, though each room has a new occupant, the number of occupied rooms is exactly the same, for eternity. Whereas before, there was an infinite number of occupied rooms and no one in the hallway, now there is the same infinite number of occupied rooms and one person in the hallway- the hallway will never be empty,

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