A supposed paradox that, unlike most suppposed paradoxes, requires you to assume a physical situation that is definitely, unmistakably impossible. So the paradox doesn't even get off the ground. However, it's there in the literature.

The mathematics is the same as in the several versions of Zeno's paradox, such as the flying arrow, or Achilles and the Tortoise, that is the addition of 1/2 + 1/4 + 1/8 + ... and never getting to the end. But Zeno didn't have the modern knowledge of how to prove that this series sums to a finite number, or perhaps he would continue to have pointed to the infinite subdivision as paradoxical anyway.

James Thomson's formulation asks you to imagine a lamp that can be switched on or off in an arbitrarily short time, and a person (or supernatural being) who can also do this with the switch. So you turn it on for one minute, then off for half a minute, then on for a quarter of a minute, then off, and so on until after two minutes you've done this infinitely many times. The question is then whether the lamp is on or off after that two minutes.

The long-winded answer, I suppose, is that you use the logic against Maxwell's demon to show that there is a lower energy limit to how small a change you can make, and there must be a Planck limit to the times it takes to change, and signals would start travelling faster than light. But all this advanced physics is just to prove that it's physically impossible... presumably to any philosophers who hadn't already noticed it was in the first sentence.

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