Present one of Zeno’s paradoxes of motion: does it deal with infinite divisibility or finite divisibility?

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amarang9 | College Teacher | (Level 2) Educator Emeritus

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If a runner is running a race, he must first go half the distance of the entire length. Before that, he must go one fourth of the entire length (half of the first half). Before that, he must travel one eighth of the full length. As this full length can be infinitely divided (half of a half of a half, ad infinitum), the runner would have to travel an infinite number of increments which seems impossible itself. Even if the smallest increment is minuscule, if there are an infinite number of them, the runner will never complete the race. Even before that can be considered, because each increment can always and forever be divided, there is no finite "first step" the runner can definitively make. If he can't move, or cover any definitive distance from the start, then what we see when the runner does run, motion, must be an illusion. This is called the dichotomy paradox because it is based on infinitely dividing into two parts. 

Another of Zeno's paradoxes of motion is the arrow paradox. At each durationless instant of time of the arrow's flight, the arrow is not moving from somewhere nor to somewhere; it is at rest. That is, at any position of its flight where it occupies space, if we froze time, we would see that the arrow is stopped and not changing position (because it is at rest). If at every one (all) of these instances the arrow is not moving from where it was nor moving to where it will be, then it doesn't move at all. It would seem, from this reasoning, that the arrow doesn't move and time is not a continuum; rather, it is made up of (durationless) instants. 

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