Zeno's Dichotomy

Zeno was a pre-Socratic philosopher and a disciple of Parmenides. They both argued that the real world isn't at all like we experience it, and Zeno argued this by presenting a series of paradoxes alleging to show that the concept of (for example) plurality led to absurdities and so there must only be one thing that exists. Many of his arguments tried to show that motion was impossible, such as the Paradox of the Arrow and Achilles and the Tortoise. But my favorite one along these lines is his Dichotomy. Take someone who runs really fast like Atalanta, a figure from Greek mythology who was so fast she ended up as a hood ornament for the Studebaker.



So Atalanta decides to run forward, say 16 meters. But obviously, before she can run 16 meters, she has to run half that distance, 8 meters. But before she runs 8 meters, she has to run half of that, 4 meters. But before that she has to run 2 meters, 1 meter, 1/2 meter, etc. The upshot is that she can never run any distance. Even if we start by saying she tries to move a Planck length forward (about 10^-35 meters), she first has to move half that distance, and half that, etc. So motion is impossible and since it seems that we and everything else moves, the world is an illusion. Roughly, the argument is:

1) Any finite distance can be divided infinitely.
2) An infinite cannot be traversed.
3) Therefore, no finite distance can be traversed.

Then along came Aristotle. He pointed out that we can use "infinite" in two different ways, as a potential amount or an actual amount. A potential infinite is an amount increasing towards infinity as a limit but never actually reaching it. That's what we symbolize with the sideways eight, ∞. At any given point, a potential infinite is a finite amount. An actual infinite, on the other hand, is not increasing towards infinity, it's achieved it. In contemporary set theory this is symbolized by aleph-null: ℵ₀. And an actual infinite amount cannot be traversed.

Now, any finite distance can be potentially divisible infinitely. But you never actually reach an "infinitieth" of the distance. So Zeno's Dichotomy, and his other paradoxes of motion, are trading on moving back and forth between the two types of infinite. This wasn't dishonest on Zeno's part, no one had made this distinction before Aristotle. In light of this distinction, though, Zeno's argument becomes:

4) Any finite distance can be potentially infinitely divided.
5) An actual infinite cannot be traversed.
6) Therefore, no finite distance can be traversed.

And this is obviously fallacious. Specifically, it commits the fallacy of the undistributed middle, where the middle term (infinity in this case) has two different definitions.

In the 19th century, Georg Cantor developed set theory which really messes with everything about infinity. So any detailed discussion of this issue has to be filtered through set theory. Nevertheless, it's still pretty interesting, no?