Imagine a scenario in which you arrive at a hotel, hot, sweaty and impatient. Your mood is not improved when the clerk tells you that they have no record of your reservation and that the hotel is full. “There is nothing I can do, I’m afraid,” he intones officiously.
If you’re in an argumentative frame of mind and know some set theory, you might in an equally officious tone inform the clerk that the problem is not that the hotel is full, but rather that it is both full and finite.
You can explain that if the hotel were full but infinite (the above-mentioned Hilbert’s Hotel Infinity), there would be something he could do. He could tell the guest in room 1 to move into room 2; the original party in room 2 he could move into room 3, the previous occupant of room 3 he could move into room 4, and so on. In general, the hotel could move the guest in room N into room (N + 1). This action would deprive no party of a room yet would vacate room 1 into which you could now move.
John Allen Paulos retelling David Hilbert’s idea