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.
QUESTION: At the end of two minutes, is the lamp on, or off?
The answer is rather startling. If we represent the successive states of the lamp by the series of increasingly short periods in which it is on and off, we obtain something that has no last member: 60, 30, 15, 7.5, 3.75, 1.875 .... The series, in other words, is infinite. So at the end of two minutes, the lamp has been switched on (and off) an infinite number of times. Now in my opinion there is nothing mathematically incoherent in the description of the experiment (unlike, possibly my analogy of half-life discussed in the last post). The sum of all the series of periods is not infinite: it approaches without quite reaching, 120 seconds.
Max Payne of the Scientific & Medical Network suggested at a recent presentation I did for the SMN that this bisection of time would come to an end when no more space and time was available for a further subdivision. He cited Cantor's Infinity Argument in support of this position. I am unaware of this and it is my intention to check this out when I have time (subjective or objective).