"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 > and no more guests can be accommodated.
That is a very poor definition of 'occupied already'
> If, conversely, the hotel is never fully occupied, there always exists > n such that n+1 is not occupied.
That is a very poor definition of 'never fully occupied'
> Either one or the other, but not both.
True, either for every room r, room r+1 is occupied OR-ELSE there exists a room r whene room r+1 is empty. not both.
But it doesn't really matter, as your definition is poor.
If you number each guest g, and put guest g in room r = 2^g, you have lots of unoccupied rooms, but every guest can be accomodated. Even infinitely many guests. And there are 'more' unoccupied rooms than occupied even when all the infinite guests are accomodated.
You could imagine this as a hotel where there is 1 room on floor 1, 2 rooms on floor 2, 4 rooms on floor 3, 2^(n-1) rooms on floor n, etc, and as each new guest arrives, they are allocated a room on a different floor.
Of course, instead of leaving empty rooms, you could just allocate rooms sequentially, and when a new guest arrive, shuffle all the existing guests to the next room.
Infinities result in some unintuitive and unexpected outcomes. Things we assume is true for finite numbers of things do not hold true for infinities. Their behaviour is well studied, however, and makes for some fascinating (if not mind-blowing) reading :D