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In philosophy, a supertask is a task occurring within a finite interval of time involving infinitely many steps (subtasks). A hypertask is a supertask with an uncountable number of subtasks. The term supertask was coined by the philosopher James F. Thomson, and the term hypertask by Clarke and Read in their paper of that name.

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 turn his argument on its head (assuming it's valid) 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 supertasks are in fact possible.

Other physically possible supertasks have been suggested. In one proposal, one person (or entity) counts upward from 1, taking an infinite amount of time, while another person observes this from a frame of reference where this occurs in a finite space of time. For the counter, this is not a supertask, but for the observer, it is. (This could theoretically occur due to time dilation, for example if the observer were falling into a black hole while observing a counter whose position is fixed relative to the singularity.)

Davies in his paper "Building Infinite Machines" concocted a device which he claims is physically possible up to infinite divisibility. It involves a machine which creates an exact replica of itself but has half its size and twice its speed.

One argument states that there should be infinitely many marbles in the jar, because at each step before t = 1 the number of marbles increases from the previous step and does so unboundedly. A second argument, however, shows that the jar is empty. Consider the following argument: if the jar is non-empty, then there must be a marble in the jar. Let us say that that marble is labeled with the number n. But at time t = 1 − 0.5^{n - 1}, the nth marble has been taken out, so marble n cannot be in the jar. This is a contradiction, so the jar must be empty. The Ross-Littlewood paradox is that here we have two seemingly perfectly good arguments with completely opposite conclusions.

Further complications are introduced by the following variant. Suppose that we follow the exact same process as above, but instead of taking out marble 1 at t = 0, one takes out marble 2. And, at t = 0.5 one takes out marble 3, at t = 0.75 marble 4, etc. Then, one can use the same logic from above to show that while at t = 1, marble 1 is still in the jar, no other marbles can be left in the jar. Similarly, one can construct scenarios where in the end, 2 marbles are left, or 17 or, of course, infinitely many. But again this is paradoxical: given that in all these variations the same number of marbles are added or taken out at each step of the way, how can the end result differ?

Some people decide to simply bite the bullet and say that apparently, the end result does depend on which marbles are taken out at each instant. However, one immediate problem with that view is that one can think of the thought experiment as one where none of the marbles are actually labeled, and thus all the above variations are simply different ways of describing the exact same process, and it seems unreasonable to say that the end result of the one actual process depends on the way we describe what happens.

Moreover, Allis and Koetsier offer the following variation on this thought experiment: at t = 0, marbles 1 to 9 are placed in the jar, but instead of taking a marble out they scribble a 0 after the 1 on the label of the first marble so that it is now labeled "10". At t = 0.5, marbles 11 to 19 are placed in the jar, and instead of taking out marble 2, a 0 is written on it, marking it as 20. The process is repeated ad infinitum. Now, notice that the end result at each step along the way of this process is the same as in the original experiment, and indeed the paradox remains: Since at every step along the way, more marbles were added, there must be infinitely marbles left at the end, yet at the same time, since every marble with number n was taken out at t = 1 − 0.5^{n - 1}, no marbles can be left at the end. However, in this experiment, no marbles are ever taken out, and so any talk about the end result 'depending' on which marbles are taken out along the way is made impossible.

A bare-naked variation that really goes straight to the heart of all of this goes as follows: at t = 0, there is one marble in the jar with the number 0 scribbled on it. At t = 0.5, the number 0 on the marble gets replaced with the number 1, at t = 0.75, the number gets changed to 2, etc. Now, no marbles are ever added to or removed from the jar, so at t = 1, there should still be exactly that one marble in the jar. However, since we always replaced the number on that marble with some other number, it should have some number n on it, and that is impossible because we know exactly when that number was replaced, and never repeated again later. In other words, we can also reason that no marble can be left at the end of this process, which is quite a paradox.

Of course, it would be wise to heed Benacerraf’s words that the states of the jars before t = 1 do not logically determine the state at t = 1. Thus, neither Ross’s or Allis’s and Koetsier’s argument for the state of the jar at t = 1 proceeds by logical means only. Therefore, some extra premise must be introduced in order to say anything about the state of the jar at t = 1. Allis and Koetsier believe such an extra premise can be provided by the physical law that the marbles have continuous space-time paths, and therefore from the fact that for each n, marble n is out of the jar for t < 1, it must follow that it must still be outside the jar at t = 1 by continuity. Thus, the contradiction, and the paradox, remains.

One obvious solution to all these conundrums and paradoxes is to say that supertasks are impossible. If supertasks are impossible, then the very assumption that all of these scenarios had some kind of 'end result' to them is mistaken, preventing all of the further reasoning (leading to the contradictions) to go through.

Similar paradoxes involving a man trying to walk a mile from A to B but the demons building a wall in front of him if he reaches 1/2, 1/4, 1/8, … of a mile past A, so he hits some sort of invisible barrier when he leaves A even though no wall has been built.

- Thomson's lamp
- Zeno's paradoxes
- Zeno machine
- NP (complexity)
- Computational complexity theory
- Omega point

- Thomson, J., 1954-55, ‘Tasks and Super-Tasks’, Analysis, XV, pp. 1-13.

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Last updated on Tuesday September 30, 2008 at 10:28:08 PDT (GMT -0700)

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Last updated on Tuesday September 30, 2008 at 10:28:08 PDT (GMT -0700)

View this article at Wikipedia.org - Edit this article at Wikipedia.org - Donate to the Wikimedia Foundation

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