Assignment 2. Zeno's Paradoxes of Motion

For submission

1. In the dichotomy, a runner must half-way along the course, and then half-way again, and so on indefinitely. If these infinite runs could be completed, the runner would have completed the source. What happens if we replace the "half" with a "third." That is, to complete the course, the runner must run a third of the way, and then a third of the way again, and so on indefinitely.

Immediately after the runner has completed all these third-point runs, where is the runner? At the end of the course? Or somewhere along the course.*

*Hint: consider the distance remaining at each stage.

2. In question 1, replace "third" with "one millionth." Again:

Immediately after the runner has completed all these millionth-point runs, where is the runner? At the end of the course? Or somewhere along the course.

3. In the Achilles, have Achilles and the tortoise switch positions. Achilles starts ahead of the tortoise and the tortoise chases after him. To catch Achilles, the tortoise must move to Achilles' starting point, and then to the position to which Achilles moved, and so on indefinitely.

In the original Achilles, this reasoning did not establish that Achilles cannot catch the tortoise. Does this reasoning now work in the switched version? If so, what has changed?

For discussion

Not for submission

A. Which of the two resolutions of the dichotomy do you prefer?

B. Some sets of infinite tasks cannot be completed. Consider one. Now consider how you would alter it to make it possible to be completed

C. For those who know calculus, is the calculus based response to the arrow better than the one in the text above?

D. The stadium does, in my view, need more sophisticated ideas for its resolution than those presented so far. Or does it? Can you resolve the paradox?