151:. Flicking the switch once turns the lamp on. Another flick will turn the lamp off. Now suppose that there is a being who is able to perform the following task: starting a timer, he turns the lamp on. At the end of one minute, he turns it off. At the end of another half minute, he turns it on again. At the end of another quarter of a minute, he turns it off. At the next eighth of a minute, he turns it on again, and he continues thus, flicking the switch each time after waiting exactly one-half the time he waited before flicking it previously. The sum of this
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It seems impossible to answer this question. It cannot be on, because I did not ever turn it on without at once turning it off. It cannot be off, because I did in the first place turn it on, and thereafter I never turned it off without at once turning it on. But the lamp must be either on or off.
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Later, he claims that even the divergence of a series does not provide information about its supertask: "The impossibility of a super-task does not depend at all on whether some vaguely-felt-to-be-associated arithmetical sequence is convergent or divergent."
276:. And this answer does not help us, since we attach no sense here to saying that the lamp is half-on. I take this to mean that there is no established method for deciding
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One of
Thomson's objectives in his original 1954 paper is to differentiate supertasks from their series analogies. He writes of the lamp and Grandi's series,
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The following question is then considered: Is the lamp on or off at two minutes? Thomson reasoned that this supertask creates a contradiction:
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this idea, just because we have the idea of a task or tasks having been performed and because we are acquainted with
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Then the question whether the lamp is on or off… is the question: What is the sum of the infinite divergent sequence
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Everywhere and
Everywhen : Adventures in Physics and Philosophy: Adventures in Physics and Philosophy
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Thomson p.6. For the mathematics and its history he cites Hardy and
Waismann's books, for which see
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Zeno, Aristotle, the
Racetrack and the Achilles: A Historical and Philosophical Investigation
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The thought experiment concerns a lamp that is toggled on and off with increasing frequency.
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207:{1, 0, 1, 0, ...}, representing the changing state of the lamp. The sequence does not
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415:. New Brunswick, NJ: Rutgers, The State University of New Jersey. pp. 209–210.
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Now mathematicians do say that this sequence has a sum; they say that its sum is ⁄
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The unending series in the brackets is exactly the same as the original series
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Infinite Pains: The
Trouble with Supertasks. In Benacerraf and his Critics
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Benacerraf, Paul (1962). "Tasks, Super-Tasks, and the Modern
Eleatics".
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Another way of illustrating this problem is to rearrange the series:
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based on infinites. It was devised in 1954 by
British philosopher
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is done when a super-task is done. … We cannot be expected to
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519:, Adam Morton and Stephen P. Stich (Eds.), p. 231-261.
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tends to infinity, so neither does the infinite series.
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199:takes the values of each of the non-negative
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469:. Oxford University Press. pp. 22–23.
172:The question is related to the behavior of
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253:that do assign Grandi's series the value ⁄
155:of time intervals is exactly two minutes.
487:(October 1954). "Tasks and Super-Tasks".
109:Learn how and when to remove this message
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45:Please improve this article by adding
515:Earman, John and Norton, John (1996)
495:(1). Analysis, Vol. 15, No. 1: 1–13.
225:= 1 − (1 − 1 + 1 − 1 + 1 − 1 + · · ·)
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409:Allen, Benjamin William (2008).
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187:= 1 − 1 + 1 − 1 + 1 − 1 + · · ·
180:the divergent infinite series
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47:secondary or tertiary sources
168:Mathematical series analogy
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380:History of Grandi's series
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436:The Journal of Philosophy
163:This is a contradiction.
308:Ross–Littlewood paradox
147:Consider a lamp with a
463:Huggett, Nick (2010).
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34:relies excessively on
656:Paradoxes of infinity
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16:Philosophical puzzle
286:transfinite numbers
191:For even values of
132:is a philosophical
596:Luigi Guido Grandi
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485:Thomson, James F.
429:on June 24, 2014.
303:List of paradoxes
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442:(24): 765–784.
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54:Find sources:
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32:This article
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318:Zeno machine
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580:Occurrences
651:Supertasks
645:Categories
403:References
99:April 2018
69:newspapers
36:references
575:Summation
570:Education
142:supertask
630:Category
489:Analysis
297:See also
209:converge
205:sequence
201:integers
605:Related
565:History
509:3326643
456:2023500
282:pick up
83:scholar
589:People
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237:= 1 −
134:puzzle
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505:JSTOR
452:JSTOR
324:Notes
90:JSTOR
76:books
471:ISBN
417:ISBN
278:what
178:i.e.
62:news
497:doi
444:doi
245:= ⁄
211:as
38:to
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243:S
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213:n
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