Countable Sets Discussion

(The following is/are the initial prompt(s) for an online discussion; students may have posted responses, and prompts for further discussion may have been added, but these things are not shown.)

This discussion helps you learn to think about countably infinite sets. It’s not as closely connected to the reading on that subject as some discussions have been, but it’s good practice. Explanations are also easily findable on the Internet, but resist the urge — it’s more fun in the long run to think about the ideas yourself before you see them explained, and there’s no reward for having a “right” answer now.

The Infinite Motel

Legend has it that tucked away in some secluded spot there is a motel with a countably infinite number of rooms. This is both a blessing and a curse for the manager: on the one hand, the motel never runs out of rooms, so the manager never has to turn guests away, but on the other hand it’s frustrating that most of the motel’s rooms stay empty night  after night. One night, however, the nearly-impossible happened: a countably infinite tour bus, completely full of tourists, pulled up, looking for a place for the tourists to stay. The motel manager was delighted — finally the Countably Infinite Motel was full! But then, just as the manage finished checking the infinite number of tourists into their rooms, a family of four pulled up in their minivan, looking for a place to spend the night.

Can the motel manager accommodate the extra four guests? Discuss what you think. If you think they can, explain how; if you think they can’t explain why not.

Now suppose that while the four new guests are either settling into their rooms, or standing forlornly in the parking lot, depending on whether or not the manager found places for them, a second countably infinite, and full, tour bus arrives. Can the manager find room for this new infinite number of guests? Discuss how they can do it, or why they can’t.