Julio Di Egidio wrote: > "Consider a hypothetical hotel with a countably infinite number of > rooms, all of which are occupied." (*) > > The nature of countable infinity is such that there cannot be a last > room, and that's fundamental to the fact that new guests can be > accommodated in an infinite hotel at will, itself an illustration of the > fact that we can count endlessly. But, if there is no last room, it can > never be the case that *all* rooms are occupied, hence the whole > argument, for how informal, falls apart since inception.
But the formal theories of infinite numbers are based on set theories, some at least of which are clearly consistent.
> 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. > > Julio > > (*) http://en.wikipedia.org/wiki/Hilbert's_paradox_of_the_Grand_Hotel > >
-- Madam Life's a piece in bloom, Death goes dogging everywhere: She's the tenant of the room, He's the ruffian on the stair.