"Julio Di Egidio" <firstname.lastname@example.org> wrote in news:email@example.com:
> "Julio Di Egidio" <firstname.lastname@example.org> wrote in message > news:email@example.com... >> >> slightly more rigorously, if we have 1, 2, 3, etc. up to (allegedly) >> *all* natural numbers, there is just no natural number missing that >> we can add to the lot. > > If, for all n, room n+1 is occupied already, the hotel is just full
Badly expressed, as your condition fails if the first room is unoccupied.
You should have said that for all n, room n is occupied.
> and no more guests can be accommodated.
That's where you're wrong when the hotel has infinite rooms
You can change the mapping from guests to rooms (ie shift the guests) and fit in one more guest. In particular move the guest in room 1 to room 2, the guest in room 2 to room 3 etc, thus makeing room 1 free for the new guest
> If, conversely, the hotel is never fully occupied, there always exists > n such that n+1 is not occupied.
Again, that fails as it ignores the possibility that the first room is not occupied
You should have said that there exists an n such that room n is not occupied.
> Either one or the other, but not both.
The problem is you assume 'fully occupied' means 'no more can be accomodated'.
Although that is the case for a hotel with a finite number of rooms, we cannot extend that to this hypothetical infinite hotel.