> "Consider a hypothetical hotel with a countably infinite number of rooms, > all of which are occupied." (*)
An anarchist enters Hilbert's Hotel with the irrationals and asks for a separate room for each. "I'm sorry," says the clerk, "We're full up, May I suggest Cantor's Hotel, up the street at omega_1?"
Immediately arrives a rationalist with the rationals, asking for a separate room for each. "No problem," says the clerk, "Please, one moment while I reassign all the guests from their current rooms to even numbered rooms twice the number of the current room. Shortly they all walk out of the lobby to their rooms.
"But, but," claims the anarchist, "you're full!" "Yes, we are," replies the clerk, "every room is occupied." "Occupied or not," demands the anarchists, "give each of us a room." "I'm sorry," repeats the clerk, "go to Cantor's Hotel where they can accommodate no counts like you."