Thank you for visiting this site. This article covers “Thomson’s Lamp” paradox.
Press a lamp’s switch once — on. Press again — off. Repeat infinitely many times. When you are done, is the lamp on or off? It turns out this question has no answer.
The Setup
British philosopher James F. Thomson introduced this thought experiment in his 1954 paper “Tasks and Super-Tasks.” His goal was to expose problems inherent in the concept of a supertask — completing infinitely many operations in a finite time.
The switch is pressed at the following moments:
- 0 seconds: switch pressed (on)
- 0.5 seconds: switch pressed (off)
- 0.75 seconds: switch pressed (on)
- 0.875 seconds: switch pressed (off)
- …
Each interval is half the previous one. Therefore, infinitely many switch operations are completed within exactly 1 second.
After exactly 1 second, is the lamp on or off?
Why There Is No Answer
Suppose the lamp is on at t = 1 second. Then the last operation was pressing it on. But every “on” press is followed by an “off” press, so there is no “last on.” Contradiction.
Suppose the lamp is off. Then the last operation was pressing it off. But every “off” press is followed by an “on” press, so there is no “last off.” Contradiction.
In other words, the concept of a “last operation” does not exist. Because the infinite sequence of presses has no final element, nothing in the setup determines the lamp’s state at t = 1 second.
Note: the claim is not that the lamp is “neither on nor off” — it must be one or the other. The point is that nothing in the setup provides any logical basis for determining which.
Difference From Zeno’s Paradox
At first glance this resembles the Achilles and the Tortoise paradox, but there is an important difference.
In Zeno’s paradox, the infinite sequence of steps converges to a finite total time, and at that moment Achilles reaches the tortoise — a well-defined outcome in a new state.
In Thomson’s Lamp, the infinite sequence also converges to 1 second, but no information in the setup determines the state at t = 1 second. The infinite operations are all completed within 1 second — but their “result” is simply undefined.
The Mathematical View
Mathematically, the function describing the lamp’s state oscillates between 0 and 1 infinitely rapidly as t approaches 1 second from below. The limit as t → 1 does not exist — the function does not converge to any single value.
This is different from Zeno’s case, where an infinite geometric series converges cleanly to a finite limit. Thomson’s Lamp produces oscillation without convergence, so no limit exists.
Therefore, “what is the lamp’s state at 1 second?” has no mathematically well-defined answer. This means the question is not so much a paradox as an ill-posed question — one the problem’s setup simply does not answer.
Benacerraf’s Reply
In 1962, philosopher Paul Benacerraf offered an influential response. His argument: the fact that the lamp’s state at t = 1 is indeterminate is not a paradox but simply a feature of the problem’s constraints.
The lamp’s state is perfectly well defined at every moment before t = 1 second. Nothing in the setup says anything about the state at t = 1 exactly. Therefore, the lamp could be on at t = 1 without contradiction, or off at t = 1 without contradiction. The fact that both are consistent with the setup is itself the answer — the problem just underdetermines the outcome.
Other Supertask Puzzles
Related supertask thought experiments exist.
The Ross–Littlewood paradox: begin with an empty vase. At each step, put 10 balls in and remove 1. Repeat infinitely. How many balls are in the vase at the end? The answer depends on which ball is removed at each step — under different removal rules the answer can be 0 or infinity.
Physically Impossible
In the real world, Thomson’s Lamp cannot be built. As the interval between presses shrinks without bound, it eventually drops below the Planck time (~5.4 × 10⁻⁴⁴ seconds), at which point the notion of a time interval loses physical meaning. The switch would also need to move faster than light.
Thomson’s Lamp is a purely mathematical and philosophical thought experiment, illustrating what happens when the concept of “infinity” is forced into a finite physical scenario.
Summary
This article covered “Thomson’s Lamp.”
Infinitely many operations are completed, yet the “result” has no definition. Perhaps demanding a “result” from an infinite process is the conceptual error itself. A deceptively simple setup that opens a window onto the abyss of infinity.
To return to the full list of paradoxes, follow the link below.
Thank you for reading. We hope to see you in the next article.
