The paradox is unsolvable, but only based on the flawed logical axiom that preforming an infinite number of tasks in a finite period of time is impossible.
Remove this idea and the problem isn't even a true paradox anymore.
Well no shit, the paradox of going back in time and killing one's own grandfather isn't a paradox if we remove the idea that you're going back in time and instead say you just moved to an alternate universe.
Being able to complete an infinite process is logically unsound without using infinitesimals, but being able to complete an infinite process is not the real problem. The real problem is human minds finding the sulution unintuitive.
The Achilles paradox is a variation on the dichotomy paradox, or "
That which is in locomotion must arrive at the half-way stage before it arrives at the goal."
Basically stating that before you can go from one point to another, you must first reach the half-way stage between those two points. And before you can get to the half-way stage you need to reach the point half-way betwen your current point and the half-way point and so on, ad infinitum.
Ergo, the conclusion is that in order to go from one point to any other point, one must first pass through an infinite number of points between the two, thus in order for an object to move, it must complete an infinite process, which is, as i previously stated, logically unsound.
From this, Zeno concludes that all motion is an illusion, i conclude that we're missing something, or possibly that distance and time are in fact discrete (if the mathematics behind Planck lengths are correct, one could put forth an argument for time and distance being
practically discrete at a Planck scale, as it's supposed to be impossible to measure anything smaller.)
Also, unrelated to the dichotomy paradox, but the Arrow paradox is actually solved by the addition of inertia.