Science of Discworld III: Darwin's Watch by Stewart Ian & Terry Pratchett & Cohen Jack

Author:Stewart, Ian & Terry Pratchett & Cohen, Jack [Stewart, Ian]
Language: eng
Format: mobi
ISBN: 9781407022697
Publisher: Random House UK
Published: 2011-06-29T16:00:00+00:00

It couldn’t stay that way.

David 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 à 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-ump gazillion 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.

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