Weeknotes S04E05:Infinities of different sizes

Do you know about Hilbert’s infinite hotel?

It’s a curious place. The room numbers start at 1, and they go on forever. There are an infinite number of rooms and tonight, they’re all full. Every single one.
A guest appears out of the darkness and asks the desk clerk if there is room. The clerk, of course, says yes. The night is dark, the guest is cold, and cold hard cash is, at the end of the day, cold hard cash. But where to put them?
It can’t be in room ‘infinity’, because that’s an absurd proposal. There’s no such number. Room 1 exists, but that room has an occupant. As does room 2. And so on, and so on.
The desk clerk considers this problem and comes to a conclusion. All that needs to be done is that everyone move up a room. The occupants of room 1 will go to room 2; room 2 to room 3, and so on. There are infinite rooms, after all. Everyone can just go to the room whose number is one greater than their current abode.

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