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Thomson's Lamp
If we start off at 0 seconds with a lamp that is off, and then turn it on at 0.5 seconds, and then turn it off at 0.25 seconds, and then turn it on at 0.125 seconds (you've got the pattern, right?) and so on till infinity...
Will the lamp be on or off at the 1 second mark?
A more involved look at Thomson's Lamp.
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Supertasks relate back to Zeno's paradox involving Achilles and the Tortoise; it is an infinite number of tasks done sequentially within a finite amount of time.
There are quite a few paradoxes that involve supertasks. Thomson's Lamp and the Ross-Littlewood paradox to name two.
