Hotel Management

Let me depart for the moment from my traditional practice of abusing Steve Landsburg to note that I quite liked his post on countable and uncountable numbers. He briefly discussed the countability of the integers and rationals and presented Cantor's proof of the uncountability of the reals. One of his commenters asked the following:

Imagine that you run the front desk at a hotel with a countably infinite number of rooms. Imagine further that all of the rooms are occupied. A man shows up in the lobby and asks for a room. Can you give him a room?

The answer is yes of course, but it should keep your bell boys busy. Interestingly enough, it's hardly any more trouble to accomodate a countably infinite number more of guests. Still, you have to say that the manager who let this deplorable situation arise made a big mistake. How should he have run his hotel so that he didn't have to move an infinite number of people each time some more guests showed up?

What if he had an uncountably infinite number of rooms (for example, let each room have a real room number between 0 and 1)? Would that have made his job any harder?

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