[T]he Hilbert Hotel doesn’t merely have hundreds of rooms — it has an

infinitenumber of them. Whenever a new guest arrives, the manager shifts the occupant of room 1 to room 2, room 2 to room 3, and so on. That frees up room 1 for the newcomer, and accommodates everyone else as well (though inconveniencing them by the move).Now suppose

infinitelymany new guests arrive, sweaty and short-tempered. No problem. The unflappable manager moves the occupant of room 1 to room 2, room 2 to room 4, room 3 to room 6, and so on. This doubling trick opens up all the odd-numbered rooms — infinitely many of them — for the new guests.

It’s the end of the From Fish to Infinity series to which I had previously linked.

How do things go if the entering convention crowd is the infinity of points on a line not the infinity of integers in rooms?

I think any infinity involving people and rooms is countable.