Zeno’s Paradox of the Arrow

Like all paradoxes, there is no contradiction here, just misuse of logic.

How do you define velocity? If you say

the distance traveled in an extended period of time, divided by that time

well then of course there's no such thing as instantaneous velocity. Asking what something's instantaneous velocity is under this definition is logically equivalent to something like

Let $n$ be the number of apples in a nonempty container of apples. What is $n$ when the container has no apples?

The question doesn't make sense, and simply cannot be answered.

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Now one can often extend definitions so that terms get defined in new circumstances, consistent with the cases for which they were previously defined. We define velocity as $$ v = \lim_{\Delta t \to 0} \frac{\Delta x}{\Delta t}. $$ This is consistent with the old in the sense that if you have a constant velocity $v$ and you travel for an extended period of time, $v$ is given by distance divided by time.

But make no mistake, our new definition goes beyond cases of extended time intervals, and in these cases the old definition still fails, just as it always did. Sure no motion occurs if no time elapses. So what? If no time elapses, the definition of velocity has nothing to do with actual distance traveled over that time.

Some object may have a nonzero velocity because our new definition of velocity says it does, whereas the old definition may have had nothing to say one way or the other. Make no mistake, the old definition does not say the velocity of an object is $0$ if no time elapses. It says the velocity of an object is currently undefined if no time elapses.