(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?
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.
But perhaps I was taught incorrectly, or I have misremembered.
In the transfinite numbers things that at first seem unintuitive happen, e.g.:
1. There are the same number of integers as there are even numbers. And there are the same number of integers as there are odd numbers.
2. There are the same number of rational numbers (fractions) as there are integers.
3. There are more real numbers than integers.