Hilbert's Hotel Paradox

HILBERT’S PARADOX

The paradox of Hilbert’s Hotel is a thought experiment about infinitely large things causing peculiar situations.

David Hilbert was a mathematician who lived almost two thousand years after Zeno, but mathematicians were still (and are still) trying to understand infinity. Hilbert’s thought experiment involves an infinite hotel, with rooms numbered 1, 2, 3, 4, and so on forever. Imagine that the hotel is full, so you also need to imagine an infinite number of people. Neither the infinite hotel nor the infinite people are possible in real life, but this is a thought experiment. Now imagine that a new guest arrives. The hotel is full, so no room is available for the new guest. However, we could move everyone up a room, so that the person in room 1 moves to room 2, the person in room 2 moves to room 3, and so on. Because there are an infinite number of rooms, everyone has a room they can move to, at the slight expense of kicking the occupant out. But after being kicked out, that occupant can in turn move to a new room. All this leaves room 1 empty and the new guest can move in.

The paradox here is not in the logic but in our intuition. In normal life, if a hotel has no empty rooms you can’t just move people around and miraculously make an empty room appear, without getting people to share rooms. The difference is that in normal life all hotels are finite. This is a veridical paradox that challenges our intuition around infinity. It warns us that we can’t just extend our intuition about finite numbers to infinite numbers, because strange things start happening. Those things aren’t wrong, they’re just different.

Hilbert’s Hotel paradox can be extended to thinking about more new guests arriving, and even infinite new guests arriving. It leads to the study of infinity as a new type of number that does not obey the same rules as ordinary numbers.

This might seem rather removed from real life as we do not have infinitely many of anything in real life. Or do we? One way we consider having infinitely many things in life goes back to Zeno’s paradox, and in the fact that any distance can be divided into infinitely many increasingly small distances. This might seem like a technicality, but remarkably this technicality enables us to understand motion and is therefore critical to everything that is automated in our modern world.

But another way we have effectively infinitely many things is by thinking about unlimited supply. Hilbert’s hotel has an unlimited supply of hotel rooms, and another vacant room can always be produced with essentially no incremental cost. This is somewhat like the situation with digital media now, as extra copies of files can be made at will, with no incremental cost. Although we do not actually have infinitely many copies of a file, effectively unlimited supply can be a useful model. Cheap copying can put pressure on prices, but does not make the content itself valueless. This is of course related to the issue of piracy, and some people argue that the supply is only infinite because of pirates, and thus that preventing people from stealing digital content will cause the supply to go back to being finite. Another point of view says that copying digital content is not really the same as “stealing”, because you’re not removing the object from someone. Indeed, the theory of infinity developed following Hilbert’s paradox tells us that subtracting one from infinity still leaves infinity. The maths can’t tell us what to do about these moral issues, but it can give us clearer terms in which to discuss them.