
Re: The selfcontradictory infinite hotel
Posted:
Feb 23, 2014 2:25 AM


On Sunday, 23 February 2014 08:23:54 UTC+2, Virgil wrote: > In article <lebotv$fc7$1@dontemail.me>, > > "Julio Di Egidio" <julio@diegidio.name> 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. > > > > One can have a bijection between two infinite sets, such as between the > > odd natuals and the even naturals. > > > > A bijection between rooms and occupants would leave no room unoccupied > > and no occupant unroomed.. > > 
Hogwash. A bijection does not lead to that conclusion at all.
A bijection exists iff a transformation is onetoone and onto. This means that f is a bijective function if f:x>y <=> finverse (y) = x for all x in the domain of f.
The catch: There is no such thing as f(oo) = k and finverse(k) = oo.

