"Julio Di Egidio" <firstname.lastname@example.org> wrote in news:email@example.com:
> "Consider a hypothetical hotel with a countably infinite number of > rooms, all of which are occupied." (*)
OK .. considering it (again)
> The nature of countable infinity is such that there cannot be a last > room,
Of course, as there is always at least one more room than any particular room you choose
> and that's fundamental to the fact that new guests can be > accommodated in an infinite hotel at will,
Yes they can.
If there is a finite number of guests, then it is obvious that any finite set of additionl guests can be accomodated, no matter how large. No matter how (finitely) many extra guests arrive, there is always room for more.
What we do is assign guests a guest-number as they arrive, and give the rooms room-numbers starting from 1
We then assign a one-to-one mapping between guest numbers (g) and room numbers (r) eg. r = g. So guest 1 is in room 1 and so on
If n extra guests arrive, we can assign them guest numbers of -n to 0, and we can change the mapping (ie shift existing guests to different rooms) eg r = g+n. The first n rooms are now assigned to the new guests.
> itself an illustration of > the fact that we can count endlessly.
We'd get a little tired, but yes, there is no maximum finite number.
> But, if there is no last room, > it can never be the case that *all* rooms are occupied,
Of course it can, if there is also no last guest.
> hence the > whole argument, for how informal, falls apart since inception.
Not really, no.
> More logical seems to say that, while there can be ideal constructs > such as the actual infinities of super-tasks and corresponding > "super-numbers", there can be no such thing as the standard countable > infinity, as that would be something that at the same time is and is > not exhausted, i.e. a self-contradictory notion. Finite infinity is > rather just that, the ever unfinished.
I'm not sure what you mean by super-tasks and super-numbers here, or what you think you mean by finite infinity.
But addressing your 'is and is not exhausted', it depends on your meaning of exhausted.
If you mean 'there are no rooms which are unoccupied' then for infinite rooms with infinite guests, that is true. If you means 'no more guests can be accomodated' then for infinite rooms with infinite guests, that is false. You need to decide what you mean by 'exhausted'.
Working with infinities can be very confusing. Some people like to simply pretend there is no such concept, but that's just hiding from the truth and imposing artificial limitation on what concepts others are allowed to have. Clearly there IS such a concept as infinities, and how they 'work' is fairly well defined (even if not necessarily intuitive).