Zeno's Achilles and the tortoise: what is the paradox?

The paradox claims that Achilles, racing a tortoise that has a head start, can never overtake it. Each time Achilles reaches the spot where the tortoise was, the tortoise has moved a little further on. The usual resolution, called the Standard Solution, says the infinitely many stages add up to a finite distance and a finite time, so Achilles does catch the tortoise and then passes it. [1] [2]

According to the Internet Encyclopedia of Philosophy, both runners move along a line at constant speeds. To catch the tortoise, Achilles must first reach where it currently is. By then it has crawled to a new location, and he must reach that one, and so on forever. Zeno concluded that anyone who believes Achilles will succeed, and who believes motion is possible at all, is the victim of an illusion. That claim had been advanced by his tutor Parmenides. [1]

Zeno's own writings do not survive. His arguments are known from his opponents, and the Achilles paradox is reconstructed from Aristotle's Physics and from passages by Simplicius in the fifth century C.E. There is no evidence that Zeno used a tortoise. Aristotle spoke only of a runner competing with Achilles, and the tortoise was a later addition by Simplicius. [1]

One teaching example gives Achilles a speed of 10 meters per second, the tortoise 5 meters per second, and the tortoise a 10 meter head start. The gap shrinks to 5 meters, then 2.5, then 1.25, halving each time, while the times needed form the sum 1 + 1/2 + 1/4 + and so on. That sum approaches 2, so Achilles draws level after exactly 2 seconds. A simpler check agrees: in 2 seconds Achilles covers 20 meters, and the tortoise, starting 10 meters ahead and running 10 meters, is also at the 20 meter mark. [2]

The source describes Zeno's hidden assumption as the belief that adding infinitely many numbers must give infinity. Treating the sum as the limit of a sequence shows that this is false for a series like this one. [2]

The Internet Encyclopedia of Philosophy says the Standard Solution grants that some uncompleted stretch of path always remains before the goal is reached, but denies that this means the goal is never reached. It treats the path as a continuum, uses calculus and classical mechanics, and treats speed as the derivative of distance with respect to time. The distances Achilles covers in each stage form a convergent geometric series whose sum is a finite distance. The record does not say which exact assumption Zeno relied on, such as too far to run, too little time, too many places, or no final step. According to the Standard Solution, all of them are false. [1]

The same source rejects two common objections. Biological limits on step size do not help, because Achilles' motion is continuous and his feet need not stop at each location. Saying Achilles should simply aim farther ahead also fails, because he must still pass through every one of those locations. [1]