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Thursday, July 8, 2010

Thomson's Lamp

"Thomson's Lamp" is an example of a "supertask," a category of paradox that involves an infinitely-divisible task. One form of the paradox runs as follows: You have a lamp that can be turned on and off with a toggle switch. At the start the lamp is turned on for exactly one minute, at which point it is turned off for exactly .5 mins., and then turned on for .25 mins., and then off for .125 mins.... and so on.

The question is, at the two-minute mark is the lamp on or off?  Common-sensically, and practically, one might expect there is a simple, correct mathematically-calculable solution to the question --- after-all, at the two minute mark the lamp MUST be EITHER on or off! But in fact, we are dealing with an infinite sequence (1 + 1/2 + 1/4 +1/8 +1/16 +....), and as such there is NO one single right answer --- different mathematical arguments/solutions can be logically made, and even semantically the problem is unsettled. In part, the answer depends on how fast one assumes the (undetailed) turning on and turning off action itself takes --- is it 'instantaneous' (requiring no amount of time), or does it take some finite amount of time (say perhaps, with the speed of light as a limiting factor)? In short, Thomson's Lamp is a fun thought exercise (involving infinity) that oddly evades any proven solution.

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