The Problem of Zeno: Part II

Last time I promised a formulation of Zeno's Paradox which would make the word "paradox" inadequate to describe the story. The most important things to consider are the definitions and assumptions. Some of these concepts are not easy to understand, so looking them up at Mathworld or on Google may prove to be useful. The key is the difference between "infinitely divisible" and "discrete".

There are many equivalent definitions of discrete, but the one I will use is that given two points in a subset of the real numbers, there are only a finite number of points between them.

Infinitely divisible is nearly the opposite of discrete, that is between any two points in a subset of the real numbers, there are an infinite number of points between them.

Now what does this have to do with Zeno's Paradox? Well everything, in my view. There is one very important concept one needs to be familiar with in order to understand this discussion and that is the idea of a function or map. This is simply an association of two points in a set, for instance if you have a map from time into space, you associate a time, like 2:00 PM Tuesday, to a place, such as a grocery store.

The tortoise implicitly assumes a few things, first that he is in one spot at any given point in time, so in terms of his mapping from time into space this means that the map is "well-defined". Since this is a race its also assumed that the participants are always moving forward, but this can be done without losing generality since even in a complicated path there will be periods where motion is only in one direction for a period of time. What this means for our hero and tortoise is that they have two maps which describe their locations during the race which are injective, that is one point in space is associated to one point in time. The tortoise also assumes that every time Achilles moves a certain period of time passes. From his argument it seems that he assumes that there is a smallest period of time which cannot be divided, which translates into the story as Achilles never catching up to him. Therefore this can be viewed as the assumption that time is discrete. With this in mind, the tortoise assumes that space is infinitely divisible since every point Achilles has to catch up to lies between his previous point and the tortoise. Now we are ready to discuss the proof of Zeno's Paradox. The fact he can always pinpoint the time Achilles reaches these points indicates the assupmtion that the map from space into time is also injective, that is there is a point in space associated to every point in time. This makes the map surjective. Note the major things revealed so far:

Time is discrete.

Space is infinitely divisible.

The map will be well-defined, injective and surjective and hence bijective.

One more definition is necessary, that of "common motion". We said nothing about our map from time into space other than it is well-defined and injective, but this definition would include all sorts of things like bouncing around the universe from one galaxy to another instantaneously. This will not do, so in order to "keep it real" so to speak, call the set of maps which represent a finite distance traversed in a finite time "common motion", i.e. something that we may see in real life. Now this includes superluminal travel, which is certainly uncommon nowadays, but for our discussion this will be a non-issue. Zeno's Paradox is now:

If space is infinitely divisible and time is discrete the set of common motion is empty, that is there are no paths which traverse a finite distance in a finite time.

Proof: Since the set common motion must consist of injective maps, we should be able to construct at least one. Assume there is a bijective map from time into space. The problem with this is that there are not enough points in time to put all the places from space. There are an infinite number of spatial coordinates which we are trying to fit in a finite number of time coordinates. This is what is known in mathematics as the "Pigeonhole principle", that is with five boxes and six oranges, if one decides to put all the oranges in the boxes, there will be at least one box with two oranges. Therefore there is no bijective map in the set of common motion. Q.E.D.

That's the argument, I hope it convinces you that under this formulation, Zeno's Paradox is indeed a theorem.