Science of Discworld III

health and safety regulations forbid more than that.)
    In Hilbert’s Hotel, however, there is always room for an extra guest. Not in room infinity, though, for there is no such room. In room one.
    But what about the poor unfortunate in room one? He gets moved to room two. The person in room two is moved to room three. And so on. The person in room umpty-ump gazillion is moved to room umpty-ump gazillion and one. The person in room umpty-umpgazillion and one is moved to room umpty-ump gazillion and two.
    The person in room n is moved to room n +1, for every number n .
    In a finite hotel with umpty-ump gazillion and one rooms, this procedure hits a snag. There is no room umpty-ump gazillion and two into which to move its inhabitant. In Hilbert’s Hotel, there is no end to the rooms, and everyone can move one place up. Once this move is completed, the hotel is once again full.
    That’s not all. On Monday, a coachload of 50 people arrives at the completely full Hilbert Hotel. No worries: the manager moves everybody up 50 places – room 1 to 51, room 2 to 52, and so on – which leaves rooms 1–50 vacant for the people off the coach.
    On Tuesday, an Infinity Tours coach arrives containing infinitely many people, helpfully numbered Al, A2, A3, …. Surely there won’t be room now? But there is. The existing guests are moved into the even-numbered rooms: room 1 moves to room 2, room 2 to room 4, room 3 to room 6, and so on. Then the odd-numbered rooms are free, and person A1 goes into room 1, A2 into room 3, A3 into room 5 … Nothing to it.
    By Wednesday, the manager is really tearing his hair out, because infinitely many Infinity Tours coaches turn up. The coaches are labelled A, B, C, … from an infinitely long alphabet, and the people in them are A1, A2, A3, …, B1, B2, B3, … C1, C2, C3, … and so on. But the manager has a brainwave. In an infinitely large corner of the infinitely large hotel parking lot, he arranges all the new guests into an infinitely large square:
    A1 A2 A3 A4 A5 …
    B1 B2 B3 B4 B5 …
    C1 C2 C3 C4 C5 …
    D1 D2 D3 D4 D5 …
    E1 E2 E3 E4 E5 …
    …
    Then he rearranges them into a single infinitely long line, in the order
    A1 - A2 B1 - A3 B2 C1 - A4 B3 C2 D1 - A5 B4 C3 D2 E1 …
    (To see the pattern, look along successive diagonals running from top right to lower left. We’ve inserted hyphens to separate these.) What most people would now do is move all the existing guests into the even-numbered rooms, and then fill up the odd rooms with new guests, in the order of the infinitely long line. That works, but there is a more elegant method, and the manager, being a mathematician, spots it immediately. He loads everybody back into a single Infinity Tours coach, filling the seats in the order of the infinitely long line. This reduces the problem to one that has already been solved. 2
    Hilbert’s Hotel tells us to be careful when making assumptions about infinity. It may not behave like a traditional finite number. If you add one to infinity, it doesn’t get bigger. If you multiply infinity by infinity, it still doesn’t get bigger. Infinity is like that. In fact, it’s easy to conclude that any sum involving infinity works out as infinity, because you can’t get anything bigger than infinity.
    That’s what everybody thought, which is fair enough if the only infinities you’ve ever encountered are potential ones, approached as a sequence of finite steps, but in principle going on for as long as you wish. But in the 1880s Cantor was thinking about actualinfinities, and he opened up a veritable Pandora’s box of ever-larger infinities. He called them transfinite numbers , and he stumbled across them when he was working in a hallowed, traditional area of analysis. It was really hard, technical stuff, and it led him into previously uncharted byways. Musing deeply on the nature of these things, Cantor became diverted from his work in his entirely respectable area of analysis, and started thinking about something much more difficult.
    Counting.
    The usual way that we introduce numbers is by teaching children to count. They learn that numbers are ‘things you use for counting’. For instance, ‘seven’ is where you get to if you start counting with ‘one’ for Sunday and stop on Saturday. So the number of days in the week is seven. But what manner of beast is seven? A word? No, because you could use the symbol 7 instead. A symbol? But then, there’s the word … anyway, in Japanese,