Supertasks

Is it possible to “complete an infinite number of tasks”?

Let $X=\{x_1,x_2,...\}$ be a countably infinite set of tasks, where each task must be completed sequentially. There are two interpretations of “complete”.

1. There exists a task $x\in X$ such that $x$ is the last task completed.

2. For all tasks $x\in X$, $x$ is completed.

Trivially, if we interpret complete in the sense of 1, then supertasks are impossible. Are supertasks possible in the sense of 2?

A common “example” of a supertask is motion. Suppose a runner moves continuously through the closed unit interval. Then, as the argument goes, the runner must arrive at 1/2, then 1/2 + 1/4, etc. We can label these as points in the interval. Then arriving at each point would be a task. The question is, does the runner ever reach 1? Obviously, he must, or else motion is impossible. But does this mean he has really completed an infinite number of tasks? 

We cannot appeal to the mathematical notion of an infinite sum, for in this case, an infinite sum is defined to be the limit of the sequence of partial sums of the series, purely conventionally. The mathematical definition of infinite summation does not assume we can sum infinitely many things. This is something many laymen, frustratingly, do not realize.

We can ask a dual question: in the motion of the runner, what is the first task that is completed? For the runner must arrive at 1/4 before arriving at 1/2, 1/8 before arriving at 1/4, etc. In other words, how does the runner even begin his motion?

We know that describing motion as a series of tasks imposes a discrete order upon motion. This is the problem with considering motion as a supertask. Motion is not a sequential completion of tasks, as the dual question above shows. Rather, motion is fundamentally distinct from discrete completion of tasks, i.e., one cannot apply the same reasoning to continuous motion that one would apply to discrete completion of tasks. I believe the fact that there is no “first” step in motion is a necessary feature of continuous motion, one that is certainly unintuitive but not paradoxical. The fact that continuous motion is fundamentally distinct from discrete completion of tasks is analogous to how the infinite is fundamentally distinct from the finite. It can certainly seem paradoxical that a set can have the same cardinality as a proper subset of itself, until one realizes that this is a fundamental property of the infinite. In much the same way, it can seem paradoxical that there is no first step in continuous motion until one realizes that this is a fundamental property of continuous motion.