A supertask is an at least countably infinite sequence of operations that occur sequentially within a finite interval of time. Supertasks are called hypertasks when the number of operations is uncountable. A hypertask that includes one task for each ordinal number is called an ultratask. The term "supertask" was coined by the philosopher James F. Thomson, who devised Thomson's lamp. The term "hypertask" derives from Clark and Read in their paper of that name. In this literature, a "task" is taken to take a finite amount of time, and at any point in time, at most one task can be in progress. This then shows that hypertasks and ultratasks are impossible, since a finite interval cannot contain an uncountable number of finite intervals. This is because a closed and bounded interval is second-countable.
History
Zeno
Motion The origin of the interest in supertasks is normally attributed to Zeno of Elea. Zeno claimed that motion was impossible. He argued as follows: suppose our burgeoning "mover", Achilles say, wishes to move from A to B. To achieve this he must traverse half the distance from A to B. To get from the midpoint of AB to B, Achilles must traverse half this distance, and so on and so forth. However many times he performs one of these "traversing" tasks, there is another one left for him to do before he arrives at B. Thus it follows, according to Zeno, that motion (travelling a non-zero distance in finite time) is a supertask. Zeno further argues that supertasks are not possible (how can this sequence be completed if for each traversing there is another one to come?). It follows that motion is impossible. Zeno's argument takes the following form:
Motion is a supertask, because the completion of motion over any set distance involves an infinite number of steps Supertasks are impossible Therefore, motion is impossible Most subsequent philosophers reject Zeno's bold conclusion in favor of common sense. Instead, they reverse the argument and take it as a proof by contradiction where the possibility of motion is taken for granted. They accept the possibility of motion and apply modus tollens (contrapositive) to Zeno's argument to reach the conclusion that either motion is not a supertask or not all supertasks are impossible.
Achilles and the tortoise Zeno himself also discusses the notion of what he calls "Achilles and the tortoise". Suppose that Achilles is the fastest runner, and moves at a speed of 1 m/s. Achilles chases a tortoise, an animal renowned for being slow, that moves at 0.1 m/s. However, the tortoise starts 0.9 metres ahead. Common sense seems to decree that Achilles will catch up with the tortoise after exactly 1 second, but Zeno argues that this is not the case. He instead suggests that Achilles must inevitably come up to the point where the tortoise has started from, but by the time he has accomplished this, the tortoise will already have moved on to another point. This continues, and every time Achilles reaches the mark where the tortoise was, the tortoise will have reached a new point that Achilles will have to catch up with; while it begins with 0.9 metres, it becomes an additional 0.09 metres, then 0.009 metres, and so on, infinitely. While these distances will grow very small, they will remain finite, while Achilles' chasing of the tortoise will become an unending supertask. Much commentary has been made on this particular paradox; many assert that it finds a loophole in common sense.
Thomson
James F. Thomson believed that motion was not a supertask, and he emphatically denied that supertasks are possible. He considered a lamp that may either be on or off. At time t = 0 the lamp is off, and the switch is flipped on at t = 1/2; after that, the switch is flipped after waiting for half the time as before. Thomson asks what is the state at t = 1, when the switch has been flipped infinitely many times. He reasons that it cannot be on because there was never a time when it was not subsequently turned off, and vice versa, and reaches a contradiction. He concludes that supertasks are impossible.
Benacerraf Paul Benacerraf believes that supertasks are at least logically possible despite Thomson's apparent contradiction. Benacerraf agrees with Thomson insofar as that the experiment he outlined does not determine the state of the lamp at t = 1. However he disagrees with Thomson that he can derive a contradiction from this, since the state of the lamp at t = 1 cannot be logically determined by the preceding states.
Modern literature Most of the modern literature comes from the descendants of Benacerraf, those who tacitly accept the possibility of supertasks. Philosophers who reject their possibility tend not to reject them on grounds such as Thomson's but because they have qualms with the notion of infinity itself. Of course there are exceptions. For example, McLaughlin claims that Thomson's lamp is inconsistent if it is analyzed with internal set theory, a variant of real analysis.
Philosophy of mathematics If supertasks are possible, then the truth or falsehood of unknown propositions of number theory, such as Goldbach's conjecture, or even undecidable propositions could be determined in a finite amount of time by a brute-force search of the set of all natural numbers. This would, however, be in contradiction with the Church–Turing thesis. Some have argued this poses a problem for intuitionism, since the intuitionist must distinguish between things that cannot in fact be proven (because they are too long or complicated; for example Boolos's "Curious Inference") but nonetheless are considered "provable", and those which are provable by infinite brute force in the above sense.
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