![]() This paradox, according to Thomson, suggests that But it would seem equally absurd to claim that it is off: forĮach moment that the lamp is turned off, there is a later moment that Lamp was turned on, there is a later moment at which it was turned It may seem absurd to claim that it is on: for each moment that the Geometric series that converges to 2 minutes, after which time theĮntire supertask has been completed. Summing each of these times gives rise to an infinite ½ a minute more we switch it off again, ¼ on, ⅛ Lamp, which he thought illustrated a sense in which supertasks truly ![]() Thomson (1954) introduced one such example now known as Thomson’s Supertasks are often described by sequences that do not converge. Which Achilles pauses for successively shorter times at each To consider the “staccato” version of the Zeno run, in Machine is that the Zeno run is continuous, while the tasks carried However, one difference between the Zeno run and a Possible for a machine to carry out an infinite number of tasks inįinite time. Zeno race is possible, then one should equally admit that it is Hermann Weyl (1949, §2.7) suggested that if one admits that the But they are not equivalent when it comes to The two meanings for the word “complete” happen toīe equivalent for finite tasks, where most of our intuitions about From Black’s argument one can see that the Zeno DichotomyĬannot be completed in the first sense. Out every step in the task, which certainly does occur in Zeno’sĭichotomy. On the other hand, “complete” can refer to carrying Since for every step in the task there is another step that happens This sense of completion does not occur in Zeno’s Dichotomy, One hand “complete” can refer to the execution of a finalĪction. On two different meanings of the word “complete.” On the Have pointed out, there is a sense in which this objection equivocates But as Thomson (1954) and Earman and Norton (1996) The existence of a final step was similarly demanded on a priori termsīy Gwiazda (2012). The Zeno task, since there is no final step in the infinite sequence. Max Black (1950) argued that it is nevertheless impossible to complete The subtleties of the choice of topology has been given by Mclaughlin Whether or not the standard topology of the real numbers provides theĪppropriate notion of convergence in this supertask. Limit as the number of steps goes to infinity. From this perspective,Īchilles actually does complete all of the supertask steps in the Textbook that deals with infinite series. Salmon (1998), and the mathematics of convergence in any real analysis Aĭiscussion of the philosophy underpinning this fact can be found in That converges to 1 in the standard topology of the real numbers. Īlthough it has infinitely many terms, this sum is a geometric series This provides a precise sense in which the following sum Mystery of Zeno’s walk is dissolved given the modern definition of a As Salmon (1998) has pointed out, much of the Modern mathematics provides ways of explaining how Achilles canĬomplete this supertask. Line (or never have started in the regressive version). Of steps cannot be completed, Achilles will never reach the finish ![]() That as a consequence, motion does not exist. Zeno, at least as portrayed in Aristotle’s Physics, argued Regressive version of the Zeno Dichotomy supertask. Getting ever-closer to the finish line (Figure 1.1.1). And he continues in this way ad infinitum, He then runs half the remaining distance again, or He then runs half of the remaining distance, or ¼ Starting line of a track and runs ½ of the distance to theįinish line. Supertasks often lack a final or initial step. 1.1 Missing final and initial steps: The Zeno walk Let us simply try to come to grips with some of the unusual mechanical Tasks, and which break down in the analysis of supertasks. Strange because our intuitions are based on experience with finite Alternatively, we might take them to seem One might take such unusual results as evidence against the Supertask was described in a 1924 lecture by David Hilbert, as Room free, and the traveler can stay the night after all. The result isĪ hotel that has gone from being completely occupied to having one The third to the fourth room, and so on all the way up. Occupant then moves to the second room, the second to the third room, Receptionist replies, “there’s plenty of space!” The first One night when the hotel is completely occupied, a traveler Strange things can happen when one carries out an infinite task.įor example, consider a hotel with a countably infinite number of
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