(An ongoing series. See part 1.)
In this capricious world nothing is more capricious than posthumous fame. One of the most notable victims of posterity's lack of judgement is the Eleatic Zeno. Having invented four arguments all immeasurably subtle and profound, the grossness of subsequent philosophers pronounced him to be a mere ingenious juggler, and his arguments to be one and all sophisms. After two thousand years of continual refutation, these sophisms were reinstated, and made the foundation of a mathematical renaissance...
—Bertrand Russell, The Principles of Mathematics (1903).
The previous post involved two different kinds of infinity. There's the infinite on, off, on, off... of Thomson's lamp. And there's the infinite division of time: one minute, then half a minute, then a quarter of a minute, etc.
Maybe your reaction to the paradox was, “Oh, that's impossible, there's no such thing as infinity in the real world.” Perhaps not. Perhaps once you get down to tiny enough fractions of a second, you see that light is emitted in tiny, discrete quanta, and to emit even one quantum of light each time the lamp turned on would require more than enough energy to burn it out.
But math refuses to make that excuse. Math deals with abstracts, ideals. Math must deal with infinity somehow. From the moment you start counting, it is always staring you in the face.
Shortly after I posted part 1, I ran across this old chat log. Oddly enough, it discusses the same two infinities, but in a different guise.
You might be relieved to know that the paradox has been resolved.
The cure was transformative.