Category: mathematics
Type: paradox
Associated thinker(s): James F. Thomson
Original publication or approximate period: Thomson, “Tasks and Super-Tasks” (Analysis, 1954)
Difficulty: Intermediate
Themes: supertasks, infinity, limits
Related experiments: Achilles and the Tortoise; Zeno’s Dichotomy; Hilbert’s Hotel
Start with a lamp off. Press its toggle button after one minute, again after another half-minute, then after another quarter-minute, and continue with each interval halved. The infinitely many presses accumulate at the two-minute mark, with no last press.
Thomson presents a dilemma: the lamp cannot be on, because every switch-on is followed by a later switch-off; it cannot be off, because every switch-off after the first is followed by a later switch-on. Yet the lamp must be one or the other. Benacerraf later challenges the inference to a contradiction: the switching instructions say what happens at each finite stage, but do not specify a separate state at exactly two minutes.
Can an infinite sequence of actions be completed in finite time, and do the instructions determine the system’s state at the endpoint?
A supertask is a task made up of infinitely many steps that is completed in a finite time. Thomson used the lamp to argue that the notion of a completed supertask is unexplained and that the described sequence cannot be completed. Benacerraf’s reply challenges the inference from the switching sequence to a contradiction; it does not prove that a physical lamp can perform the sequence.
The case separates two questions: what the instructions specify at every finite stage, and what—if anything—they specify at the limit time.
Thomson’s argument treats the incompatible-seeming end-state reasoning as evidence against completing the task. Benacerraf’s reply is that either “on” or “off” can be assigned at the terminal instant without contradicting the switching instructions alone. That reply does not show that a supertask is physically possible; it says the described sequence does not itself settle the endpoint.
The alternating sequence of binary lamp states has no ordinary real-valued limit. The SEP discussion notes that Cesàro averaging can assign it a mean of one-half; that mean does not tell us whether the binary lamp is on or off. Physical implementations raise additional questions beyond this mathematical description.
Saying that an actual lamp would burn out changes the idealized setup rather than resolving its question. The lamp is also different from Zeno’s Dichotomy: the latter uses a convergent sequence of distances, while the lamp’s alternating binary states have no ordinary limit. A convergent total time for the intervals does not, by itself, assign a lamp state at the endpoint.
The Ross–Littlewood urn is another supertask, but it uses a different sequence of operations and raises a different question about the final state.
The relevant lamp discussion in the Stanford Encyclopedia of Philosophy entry on “Supertasks,” §1.2, was read alongside selected passages in Thomson’s 1954 paper (pp. 5–8 and 12–13) and Benacerraf’s 1962 paper (pp. 768–774 and 779–781). Thomson’s paper was read in a JSTOR scan hosted by the London School of Economics; Benacerraf’s selected text was read in a third-party hosted PDF, while the publisher record was used for bibliographic metadata. The complete primary articles were not read systematically. The entry reports the disagreement about the endpoint without claiming that either argument settles whether a supertask is physically possible.
Research check: 2026-11-01. This is a summary of the published arguments, not a claim that a supertask is possible or impossible.