So, I've come across this paradox recently, and I have not been able to figure it out, although I'll include my best answer. I would like to see what you guys think! Here goes:
So, Achilles is having a race with a tortoise, Achilles is obviously much faster, so the tortoise is given a 100 unit head start. Achilles quickly sprints to where the tortoise was, moving 100 units forward, while the tortoise has moved 10 units in the same time. In the next segment of the race (occurring in a tenth of the times), Achilles sprints 10 units forward while the tortoise has moved 1 unit. The paradox occurs when you repeat this process and conclude that the tortoise will always be in front, even though Achilles in 10x faster.
My best answer:
The sequence is being interpreted as a geometric(?) progression in which the tortoise will always win, though in reality it is arithmetic, meaning that the tortoise's movement is geometrically independent from that of Achilles, and vice versa. According to this, Achilles will not move 10x the speed of the tortoise, but at a pace of 100 units / segment, while the tortoise moves at a speed of 10 units / segment + 100 units. From this I can gather that Achilles will surpass the tortoise as an indeterminable distance of X from the point of origin if X is used in the inequality 110<X<112., X being infinitely close to 112 units from the origin of the race. (Or, X=111.111...). This statement also states that if, as posed by Zeno's paradox, Achilles never surpasses the hare, the race will have to end at a point before X. This, however cannot be true, since the race is an infinite distance and the racers are infinitely traveling geometric points.
