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This and other paradoxes of Zeno are put forward by him in the framework of a polemic about the fundamental concepts of physics and mathematics. Thanks to the school, we now understand the basic things, but then a lot of things that are obvious today were not obvious, and there were fierce disputes around. It was only with Aristotle's “Metaphysics” that the general ideas about what motion, time and space are were more or less established, and these ideas, despite their factual inaccuracy, existed right up to Galileo, until he experimentally refuted many of them.
Specifically, the paradox of Achilles and the tortoise is intended to be an argument against the possibility of infinitely dividing time and space, according to Zeno, if this were possible, then Achilles could not catch up with the tortoise. This paradox may seem very absurd to us, but we should not forget that Greek mathematics was very primitive and they did not know many modern concepts. For people who have stopped their development at the level of elementary school knowledge, their arguments are strikingly elegant and complex.
This fact becomes “obvious” only after conducting an experiment, or adding such a concept as “speed”to the data. In the proposed paradox, only 2 variables are considered: distance and time. If you have carefully read/listened to this paradox, it only says that in the time it takes Achilles to reach the place where the turtle was a few moments ago, the unfortunate animal will crawl away some more distance. At some point in time, the distance to which the turtle will crawl away will become negligible, but even in this case, without introducing the concept of “speed”, it is impossible to refute the statement stated in the paradox.
Not at all, because for mathematics of that time in aporia, the distance between them was infinitely reduced, and did not converge to zero for a number of reasons, which of course can now be refuted.�
For you, of course, Achilles will catch up, but you don't build evidence for this – you just draw a logical conclusion. But you can't always do this, because then you can say that the hypotenuse will logically ever come into contact with the asymptote, that the series always has a sum, that the root of a negative number cannot be found, that heavy bodies always fall faster than light ones, that the earth is flat …
Zeno, in fact, replaced the infinity of time with the infinity of dividing into smaller and smaller pieces the same finite period of time, which does not include the moment of catching up with the turtle (but is limited to this moment on the one hand and the moment of starting – on the other). Simultaneously with time, Zeno also artificially limited the distance under consideration.
Within the framework of the aporia itself, you can prove its inconsistency by finding the distance at which Achilles will catch up with the turtle. It is found as the (finite) sum of an infinite number of elements of a geometric progression with a denominator less than one.
If you additionally find out the speed of Achilles, you can also determine the (final) time by dividing the distance by the speed. If the speed value is unknown, it is enough to simply limit ourselves to the formula and make sure that the time will be finite at any non-zero speed.
I don't know how much Zeno was aware of the cheating nature of his aporia.