sharpens them. Mathematicians worked out that some infinities are harmless, others are not.
The only problem left after that brilliant insight was: how do you tell? The answer is that if your concept of infinity does not lead to logical contradictions, then it's safe to use, but if it does, then it isn't. Your task is to give a sensible meaning to whatever `infinity' intrigues you. You can't just assume that it automatically makes sense.
Throughout the eighteenth and early nineteenth centuries, mathematics developed many notions of `infinity', all of them potential. In projective geometry, the `point at infinity' was where two parallel lines met: the trick was to draw them in perspective, like railway lines heading off towards the horizon, in which case they appear to meet on the horizon. But if the trains are running on a plane, the horizon is infinitely far away and it isn't actually part of the plane at all - it's an optical illusion. So the point `at' infinity is determined by the process of travelling along the train tracks indefinitely. The train never actually gets there. In algebraic geometry a circle ended up being defined as `a conic section that passes through the two imaginary circular points at infinity', which sure puts a pair of compasses in their place.
There was an overall consensus among mathematicians, and it boiled down to this. Whenever you use the term `infinity' you are really thinking about a process. If that process generates some well determined result, by however convoluted an interpretation you wish, then that result gives meaning to your use of the word `infinity', in that particular context.
Infinity is a context-dependent process. It is potential.
It couldn't stay that way.
avid Hilbert was one of the top two mathematicians in the world at the end of the nineteenth century, and he was one of the great enthusiasts for a new approach to the infinite, in which - contrary to what we've just told you - infinity is treated as a thing, not as a process. The new approach was the brainchild of Georg Cantor, a German mathematician whose work led him into territory that was fraught with logical snares. The whole area was a confused mess for about a century (nothing new there, then). Eventually he decided to sort it out for good and all by burrowing downwards rather than building ever upwards, and putting in those previously non-existent foundations. He wasn't the only person doing this, but he was among the more radical ones. He succeeded in sorting out the area that drove him to these lengths, but only at the expense of causing considerable trouble elsewhere.
Many mathematicians detested Cantor's ideas, but Hilbert loved them, and defended them vigorously. `No one,' he declaimed, `shall expel us from the paradise that Cantor has created.' It is, to be sure, as much paradox as paradise. Hilbert explained some of the paradoxical properties of infinity a la Cantor in terms of a fictitious hotel, now known as Hilbert's Hotel.
Hilbert's Hotel has infinitely many rooms. They are numbered 1, 2, 3, 4 and so on indefinitely. It is an instance of actual infinity - every room exists now, they're not still building room umpty-ump gazillion and one. And when you arrive there, on Sunday morning, every room is occupied.
In a finite hotel, even with umpty-ump gazillion and one rooms, you're in trouble. No amount of moving people around can create an extra room. (To keep it simple, assume no sharing: each room has exactly one occupant, and 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 A1, 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: Al 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 El ...
(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.[1]
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 actual
[1] If you've never encountered the mathematical joke, here it is. Problem 1: a kettle is hanging on a peg. Describe the sequence of events needed to make a pot of tea. Answer: take the kettle off the peg, put it in the sink, turn on the tap, wait till the kettle fills with water, turn the tap off ... and so on. Problem 2: a kettle is sitting in the sink. Describe the sequence of events needed to make a pot of tea. Answer: not `turn on the tap, wait till the kettle fills with water, turn the tap off ... and so on'. Instead: take the kettle out of the sink and hang it on the peg, then proceed as before. This reduces the problem to one that has already been solved. (Of course the first step puts it back in the sink - that's why it's a joke.)
infinities, and he opened up a veritable Pandora's box of ever-larger infinities. He called them trans-finite 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.