How Zeno’s Paradox is a paradox?

Question

For those of you who aren’t familiar: Achilles and a Tortoise race, however the tortoise is given a leading start. Achilles is at Point A, whereas the tortoise is ahead at point B. The race begins, and by the time Achilles makes it to point B, where the Tortoise used to be, it has reached point C. Then Achilles arrives at point C with the Tortoise at point D. So on and so forth, with Achilles never catching up to the Tortoise as per the paradox.

But he definitely catches the Tortoise eventually, right? The tortoise has a lower velocity, hence the head start, so after a certain amount of time the distance between points is smaller than Achilles and the Tortoise’s difference in speed. What, if anything, is paradoxical about the world’s most famous paradox?

In: 9

Answer 1

Imagine that I gave you a task:

“Start counting up in integers (1, 2, 3, etc.). When you finish counting all the integers then you can have a slice of cake.”

When are you going to have your slice of cake? You are never going to run out of integers since you can keep counting up for an infinite period of time, so in theory you should never reach the point where you can have the cake, right?

Now imagine that as you are counting the time between 1 and 2 takes a second, but you start speeding up so that the time from 2 to 3 is only a half second, from 3 to 4 a quarter of a second, etc. Conceptually this doesn’t matter since we didn’t really care about how quickly you were counting in the first example, as the issue of the integers being infinite was the real issue. But in this case somehow you “finish” and eat the cake?

This is the idea behind Zeno’s Paradox. In order for the hare to pass the tortoise it must first reach the tortoise, and in order to do that it must reach half the distance between it and the tortoise. If it reaches the halfway point then it must next reach the new halfway point, and when it does that reach the *new* new halfway point, etc. In concept this cycle can be continued infinitely since distance is infinitely divisible and so there are equally an infinite number of steps to this task as there are integers in the previous examples. Yet in this case we know the hare will be able to pass the tortoise. Somehow an infinite series of tasks was completed in finite time.