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"THOMPSON'S LAMP: Consider a lamp, with a switch. Hit the switch once, it turns it on. Hit it again, it turns it off. Let us imagine there is a being with supernatural powers who likes to play with this lamp as follows. First, he turns it on. At the end of one minute, he turns it off. At the end of half a minute, he turns it on again. At the end of a quarter of a minute, he turns it off. In one eighth of a minute, he turns it on again. And so on, hitting the switch each time after waiting exactly one-half the time he waited before hitting it the last time. Applying the above discussion, it is easy to see that all these infinitely many time intervals add up to exactly two minutes."

"QUESTION: At the end of two minutes, is the lamp on, or off?"

Conceptually, without mathematical background, I would say ON since it was ON when it began. But why? Why not? Mathematically?

"ANOTHER QUESTION: Here the lamp started out being off. Would it have made any difference if it had started out being on?"

Again, conceptually, I would say yes, it would be ON if it started ON. But what's the answer? Mathematically?

Thanks.

David