1. If it is possible for Achilles to move from point A to point B,
then Achilles can traverse infinitely many intervals.
2. Achilles cannot traverse infinitely many intervals.
3. Therefore, it is not possible for Achilles to move from point A to point B.

This is a famous paradox that tries to prove that motion is impossible. In the form presented, it is actually an infinite class of paradoxes branching on premise (2) as their subconclusion. We will not resolve the infinite class, but instead lay to rest two popular arguments within the class.

#### Infinite Time Subargument

1. It takes a finite amount of time to traverse any interval.
2. So, it takes an infinite amount of time to traverse an infinite number of intervals.
3. If it is possible for Achilles to traverse infinitely many intervals, then
Achilles must take an infinite amount of time.
4. Achilles cannot take an infinite amount of time.
5. Therefore, Achilles cannot traverse infinitely many intervals.

Here, premise (2) is not true. It may be expanded as a subargument making its own infinite class of arguments, but this time it is easier to notice that time converges to a finite value just as distance does because the time it takes to traverse each interval is proportional to the length of the interval: Say t=time for interval, l=interval length, L=total distance. Then t=kl for some constant k. Sum[t]=Sum[kl]=k*Sum[l]=k*L which is finite.

1. The conclusion of Thomson's Lamp argument is true.
2. If the conclusion of Thomson's Lamp argument is true, then it is impossible
The property that a continuous-task has that a general task does not is the following: as the time taken for the task goes to zero, the quantity of change to the state of the universe goes to zero. A discete-task has the property that it does not satisfy this requirement, so both type are different than tasks in general and different from each other by Leibnitz's Law. As a side note, discrete-tasks and continuous-tasks, form a partition of all tasks so that any tasks can be classified as one or the other. Switching Thomson's Lamp is a discrete-task because no matter how fast the switch is flipped, it always causes the same change in the state of the universe; we could say that the number of lamps turned on changes by 1. Traversing an interval is a continuous-task because if you look at a short enough interval of time, you can force the distance traversed to be smaller than any given value (this of course assumes a finite velocity). To illustrate this concept better, we may create task-functions to represent the state of the universe versus time. The task-functions for the lamp and motion are below: 