MATH EXPLORERS' CLUB Cornell Department of Mathematics

# Puzzles and Paradoxes: Infinity in Finite Terms

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## Difficulties: Infinity Machines

We've seen how useful and productive the concept of infinite processes can be. I would like to close by discussing two interesting cases where, try as we might, we can't seem to make sense of the `final' result of an infinite process at all.

The first case is known as Thomson's Lamp, named after James F. Thomson who was the first to write about it. We imagine a special kind of machine hooked up to a desk lamp. The sole function of the machine is to flick the lamp on and off; it is special because of the speed with which it can do this. Much like a 100 meter runner is able to complete infinitely many tasks in a finite period of time by accomplishing each one successively faster, our machine is able to complete infinitely many flicks in a finite period of time.

It starts by flicking the lamp on, which takes 1 second. Half a second later, it flicks the lamp off. A quarter of a second after that, it flicks the lamp back on again, and it continues according to this pattern. According to our work on infinite geometric series, we calculate that the machine will be 'done' in exactly (1 + 1/2 + 1/4 + ...) = 2 seconds. So far, so good.

Here's the difficulty: once the machine is done, take a look at the lamp. Is it on, or is it off?

Because of the way the sequence of flicks was set up, there is no 'last flick' despite the fact that the whole process stops after two seconds. Compare this to Zeno's paradox, where there is no 'last leg' of the infintely many legs of the race, even though it's all over in 10 seconds. For our runner, however, there's no need to consider the last leg of the race, since we know he ends up at the finish line and nothing remains to question.

The lamp is different: we don't just want to know that the machine stops, we also want to know what state the lamp is in when it's all over! Yet there seems no reason for the lamp to be in either state rather than the other. What went wrong here?

Opinions are divided when it comes to this question. One easy answer is just that such a machine is impossible: there's a maximum speed at which a light can be flicked on and off, so it can't happen arbitrarily quickly. This seems reasonable, but it's not clear that it is truly satisfying. Who's to say that someone, somewhere, won't find some other sort of task that is possible to do arbitrarily quickly? The problem would then recur without such a simple solution.

Machines such as Thomson's lamp-flicking machine that can complete certain tasks arbitrarily quickly are sometimes referred to as infinity machines. In the lamp example, the existence of such an infininity machine would imply something strange about the world: that there are some present states of affairs (like a lamp currently being on or off) that are not determined by the past; they're 'free floating' in the flow of causation.

While this is a strange thought, it's not altogether without sense or even precedent. But infinity machines can actually throw much bigger monkey wrenches into the works of the universe, as we'll now discover.

Let's return to the case of the runner all set for the 100 meter dash. This time, however, some trickster has built an infinity machine and attached it to the track. It works as follows: as soon as the runner travels a positive distance d > 0 past the start line, the machine senses this and immediately puts up a force field ahead of the runner at a distance of 2d from the start line. So, for example, when the runner gets to the 20 meter mark, the machine will put up a force field at the 40 meter mark, thereby preventing the runner from ever going past 40 meters.

On the other hand, the runner can't actually ever get to the 20 meter mark, since before he gets there he'll have to pass, for instance, the 5 meter mark, at which point a force field will spring up at the 10 meter mark and prevent any progess past that point.

In fact, I claim that the runner can't even start the race. To see this, suppose that the runner does start the race. Then he must travel some distance past the start line, say d meters. But before he gets there, he will have to pass the d/3 meter mark, thus causing a force field to spring up at the 2d/3 meter mark and preventing him from ever having gotten to the d meter mark in the first place!

So it's impossible for the runner to even start the race with the infinity machine attached to the track. However -- and here's the main point -- if the runner doesn't start the race, and so doesn't travel any distance at all, then the machine never activates. There might as well not have been a machine there at all (since it never actually creates a single force field). Right? But then -- why can't the runner start?

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