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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 of π in finite time. It would simply have to print the first digit of the decimal expansion after ½ minute, the second after ¼ minute, and so on. It would thus have printed an infinite number of digits after 1 minute had elapsed.

That could open the floodgates to all kinds of astonishing things. Alan Turing, pioneer of computing, showed that there exist mathematical operations that cannot be carried out by any computer in a finite number of computational steps. They are called uncomputable operations, and their existence is closely associated with the famous incompleteness theorem of the mathematician Kurt Gödel. This teaches us that there exist arithmetical propositions that we can never prove to be true or false by using the rules of arithmetic.

Uncomputable tasks require more than just performing the same trick over and over again; they demand something novel to be introduced at each step. Their defining characteristic is that a normal computer set the task of carrying them out would never stop. Mathematicians have already identified many such operations and proved that they are uncomputable. However, an infinity machine could solve uncomputable problems in finite time. So if supertasks are possible then the consequences are dramatic and far reaching.

But back to the question: is the light on or off? Maybe it depends who you ask. A physicist or engineer would tell you it is physically impossible for such a sequence of switchings to take place. Quantum mechanics does not allow us to measure energies and time simultaneously with arbitrary precision, and the nature of space and time fluctuates wildly on intervals less than 10^-43 of a second. And even if the demon could control his switching with this precision, he would not be able to move at the ever-increasing speeds needed to press the button an infinite number of times in 1 minute.

And if you ask a mathematician? Well, infinite sequences can have bewildering properties. Suppose that we consider an unending sequence of switchings scoring +1 for on and -1 for off. Then by adding up the alternating sequence of +1 and -1 scores we can determine whether the lamp is off or on after any finite number of switchings.

If we start with the lamp off then the sequence of on-off switchings gives us a sum that looks like

1 – 1 + 1 – 1 + 1 – 1 + …

After any even number of switchings the total will be zero and the light will be off; but after any odd number of switchings the total will be +1 and the light will be on.

So, if we want to know whether the light is on or off after an infinite number of switchings all we have to do is find the infinite sum of the series. However, we can group or rearrange the terms in the series so that the sum is either 0 or 1 (consider (1 – 1) + (1 – 1) + … and 1 + (1 – 1) + (1 – 1) + …). Even stranger, we can break the series up as

S = 1 – (1 – 1 + 1 – 1 + 1 – 1 + …)

Nothing odd about that, but notice that the unending series in the brackets is exactly the same as our original series S; so we have

S = 1 – S

which implies S = ½: the lamp is neither off nor on, it is in a state that is like an average of the two.

The contradiction arises because this infinite series has no unique sum. We can only define its sum if we also specify the process used to enumerate it. This is unique to infinity: it is never the case for the sum of a finite number of terms. Thus the question of whether the lamp is on or off is meaningless. Sorry.