In Hilbert's paradox to accommodate a new guest you move guest 1 to room 2, guest 2 to room 3 and so on this will make space for a new guest. I assume that guest 1 has to wait for guest 2 to move to room 3 before he can move to room 2 thus all guests have to wait for guest n to move to room n+1. My question is that since there are infinite rooms the procedure of moving guests is infinite so does that mean that a room will never be made available?

Curious3141
Homework Helper
In Hilbert's paradox to accommodate a new guest you move guest 1 to room 2, guest 2 to room 3 and so on this will make space for a new guest. I assume that guest 1 has to wait for guest 2 to move to room 3 before he can move to room 2 thus all guests have to wait for guest n to move to room n+1. My question is that since there are infinite rooms the procedure of moving guests is infinite so does that mean that a room will never be made available?
Can't the hotel manager just yell out, "Attention!!! All guests out into the corridor! Quick march at my command to the room to your right - 1-2-3!!!"? :rofl:

So the move can be synchronised perfectly without anyone needing to wait.

You must also realise that this is purely a thought experiment, so practical considerations become meaningless. If you do manage to build a hotel with $\aleph_0$ rooms, let us know.