Supertasks: Doing the Impossible, Infinitely Fast

You can do the impossible. You just have to perform a supertask. A supertask is the completion of infinitely many steps in a finite amount of time. It sounds absurd. But whether it is truly impossible is one of the most fascinating open questions in the philosophy of infinity.

What Is a Supertask?

The term was coined by the philosopher James F. Thomson in 1954. A supertask occurs when an agent performs infinitely many actions in a finite interval. The simplest example: perform the first action in 1/2 minute, the second in 1/4 minute, the third in 1/8 minute, and so on. The total time is 1/2 + 1/4 + 1/8 + ... = 1 minute. After one minute, you have completed infinitely many actions.

The mathematical convergence is not in dispute. The question is whether the scenario is physically and philosophically coherent.

Thomson's Lamp

Thomson devised a famous puzzle. You have a lamp with a toggle switch. At the start, the lamp is off. After 1/2 minute, you turn it on. After 1/4 minute more, you turn it off. After 1/8 minute more, you turn it on. And so on. After 1 minute, you have toggled the switch infinitely many times.

Is the lamp on or off at the end?

It seems like it must be one or the other. But the sequence of states is off, on, off, on, off, on, ... with no final term. The sequence does not converge. There is no mathematical answer, because the function is not defined at the limit point.

Thomson argued that this shows supertasks are logically impossible. Others (notably Paul Benacerraf in 1962) argued that it only shows the sequence does not determine the final state, not that no final state is possible. Perhaps the lamp is on, or perhaps it is off, but nothing in the description of the supertask forces either answer.

The Ross-Littlewood Paradox

An even stranger puzzle: you have an empty vase and infinitely many numbered balls. At step 1 (at 1/2 minute before noon), you put balls 1-10 in the vase and remove ball 1. At step 2 (at 1/4 minute before noon), you put balls 11-20 in the vase and remove ball 2. At step 3, you put balls 21-30 in and remove ball 3. And so on.

At each step, you add 10 balls and remove 1, so the number of balls in the vase increases by 9 each step: 9, 18, 27, ... This sequence diverges to infinity. So at noon, the vase should contain infinitely many balls.

But which balls are in the vase? Ball 1 was removed at step 1. Ball 2 was removed at step 2. Ball n was removed at step n. Every ball was removed at some point. So the vase is empty.

The vase contains infinitely many balls (by counting) and zero balls (by tracking individuals). The paradox reveals that our intuitions about infinity are deeply unreliable when applied to completed infinite processes.

Supertasks and Zeno

Supertasks connect directly to Zeno's paradoxes. If Achilles must cross infinitely many intervals to catch the tortoise, then catching the tortoise is a supertask. The mathematical resolution (the series converges) shows that the supertask is completed in finite time. But the philosophical question remains: is the completion of infinitely many steps genuinely possible, or is the mathematical convergence a formal trick that papers over a real impossibility?

This is closely connected to the question of actual vs. potential infinity. If only potential infinity is real (Aristotle's view), then supertasks are impossible, because they require the completion of an actual infinity of steps. If actual infinity is real (Cantor's view), then supertasks may be possible in principle, even if we cannot perform them in practice.

Physical Supertasks

Some physicists have investigated whether the laws of physics permit supertasks. In certain spacetimes allowed by general relativity (so-called Malament-Hogarth spacetimes), it is theoretically possible for one observer to witness another observer completing infinitely many computations. Whether such spacetimes describe our actual universe is unknown.

Suggested Reading

James F. Thomson, "Tasks and Super-Tasks" (Analysis, 1954). The original paper.

Paul Benacerraf, "Tasks, Super-Tasks, and the Modern Eleatics" (Journal of Philosophy, 1962). The influential reply to Thomson.

Jon Perez Laraudogoitia, "Supertasks" (Stanford Encyclopedia of Philosophy). A comprehensive survey of the topic.

A.W. Moore, "The Infinite" (Routledge). Chapters on Zeno, Aristotle, and the completion of infinite processes.