26 January 2005

Supertasking with the genie of the lamp

ONE example of a possible supertask, invented in 1954 by the philosopher James Thomson, asks us to imagine a reading lamp with an on-off switch. Suppose the light is off to start with. If you press the button once, or any odd number of times, the lamp will be on. Press it an even number of times and the lamp will be off.

A little demon now appears and decides that he will keep pressing the button so as to leave the lamp on for ½ minute, then off for ¼ minute, on for 1/8 minute, off for 1/16 minute and so on. Simple mathematics shows us he will have pressed the button an infinite number of times after 1 minute. So the question is: will the light be on or off once he has finished?

It’s more than a philosophical question. A machine able to complete a supertask like this could determine the entire infinite decimal expansion

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