The Supertask

In Greek mythology, Achilles was a hero of the Trojan War. Some 2500 years ago, philosopher Zeno of Elea included him into one of his paradoxes. Achilles was quite the runner... or maybe the paradox shows otherwise.

Suppose Achilles was in a 100 meter foot race with a tortoise. The tortoise is given a 25 meter head start. It would seem obvious that when Achilles eventually starts running, he will easily overtake the tortoise and then comfortably reach the finish line (with time to spare). Looking at it logically though, one could wonder how Achilles can ever beat the tortoise.

When Achilles starts, he first has to run the distance the tortoise has already covered; in this case 25 meters. But while this happens, the tortoise won't sit still. Let's say he is able to move another 30 centimeters within this time. So after 25 meters, Achilles still has to run those extra 30 centimeters too. And while he does that... well you catch my drift. Each time Achilles covers the distance to the tortoise, there's an additional distance the tortoise has moved that Achilles still has to catch up to. And whilst this distance will quickly become infinitely small, it's there. It seems overtaking would never be possible.

Is there a flaw in the logic? How does it hold up to our common sense? Do these infinitely small distances have any meaning in the real world? After two and a half millennia, the fields of mathematics, philosophy and physics have yet to agree on a mutual solution.

Zeno's paradoxes describe an infinite amount of actions that need to occur in a finite amount of time. Those are called supertasks. Consider another.

It's a thought experiment and I'll let you do it — here's a lamp with a toggle switch. I have a timer and when I start it, you turn the lamp on. After a minute, you turn the lamp off. Then after half a minute, you turn it back on again. You'll wait a quarter of a minute and... that's right, you turn the lamp off. Continue like this: each time flick the switch after waiting exactly half the time you did before. Pretend there's no limit to how fast you can do this. Infinitely fast and infinitely many times, actually. Ultimately this experiment will take exactly two minutes, as this is the sum of all those timer intervals. So after 2 minutes, is the lamp on or off?

Heh. Maybe there's no answer. Surely, there are problems with the physics of the whole thing. Contrary to Zeno's supertask, it can't be done in reality. But how about logically? Well, let's say the lamp was on after 2 minutes. This means that immediately before the 2-minute mark you've switched it on. Yet after you switch it on, you switch it off every time. That's what you do! And you do this halfway between the time you've just switched it on, and the 2-minute mark. So the lamp can never end up being on. Of course, I can say exactly the same thing (in reverse) about the lamp ending up off. It's a contradiction...!

I'm afraid I have no satisfying resolution for you here. All I can point out is this: it can be determined whether the lamp is switched on or off for all times under the exact 2-minute mark. Since it is never stated how this on/off-switching sequence ends, one can never tell what the last flick of a switch is. It's indeterminate. Will this do?