Here are some interesting questions on the Hilbert hotel which boggles my mind. Shortly, If you can keep as many rooms as you wish unoccupied, can you still claim your hotel is full?
(1) Example, if you can move everyone from N to N+1 or (N+5); and get an unoccupied room for new reservations, can you still claim anyone that your hotel with infinite rooms is fully occupied?
(2) Or is a fully occupied infinite-roomed hotel an oxymoron, or a self-contradictory dasein; which can not exist upfront?
(3) Or is it better to call an infinite-roomed hotel both fully occupied and fully available, a superposition of both predicates, or un-predicatable?
I think the mathematical definition of 'the hotel is full' would be 'there is a one-to-one correspondence, also called an isomorphism, between the hotel rooms and occupants'.
So, if the hotel has empty rooms, even if there are an infinite number of occupied rooms it is not 'full'.
On a side note: no matter how many occupants there are in the hotel one might argue that it was also 'empty' given that it can always accommodate a countably infinite number of new arrivals.
(1) Example, if you can move everyone from N to N+1 or (N+5); and get an unoccupied room for new reservations, can you still claim anyone that your hotel with infinite rooms is fully occupied?
(2) Or is a fully occupied infinite-roomed hotel an oxymoron, or a self-contradictory dasein; which can not exist upfront?
(3) Or is it better to call an infinite-roomed hotel both fully occupied and fully available, a superposition of both predicates, or un-predicatable?