Zeno's Paradox is the name given to the most famous of a series of paradoxes put forth by Zeno of Elea, sometime in the 400's BC. It is the motion paradox sometimes known as "Achilles and the Tortoise''. The problem arose as a critique against the idea (gaining popularity at the time) that motion is continuous, or infinitely divisible. Zeno constructed a simple example -- the Roman warrior Achilles chasing a tortoise -- which, along with an assumption of infinitely divisible motion, results in the apparent conclusion that Achilles never overtakes the tortoise despite his supposed greater speed. Zeno's argument is an example of reductio ad absurdum, in which a statement is proven false (the infinite divisibility of motion) by showing that its logical consequences are untenable (that a faster object never overtakes a slower one.) But Achilles does overtake the tortoise, as everyday experience confirms. Are we to conclude that motion is therefore not infinitely divisible? Not so fast. Even if motion is perfectly continuous and infinitely divisible, Achilles will reach the tortoise. Zeno was wrong! Let's see why.

First, we need to lay out Zeno's argument in more detail. Suppose that the initial position of Achilles is given by $${\color{black}{A_0}}$$, and that of the tortoise by $${\color{black}{T_0}}$$ (see Figure 1). Further, suppose that Achilles and the tortoise are both moving to the right, with Achilles moving at twice the speed of the tortoise (the ratio of speeds is not important, so long as Achilles is moving faster than the tortoise.) Some time later, Achilles will reach the point $${\color{black}{T_0}}$$ -- the starting position of the tortoise -- but by this time the tortoise will have moved to $${\color{black}{T_1}}$$ (in our example, he's gone half as far as Achilles). When Achilles later reaches the point $${\color{black}{T_1}}$$, the tortoise has again inched forward, to $${\color{black}{T_2}}$$. If motion is continuous, this proceeds ad infinitum: whenever Achilles reaches the point $${\color{black}{T_n}}$$ for any $${\color{black}{n > 0}}$$, the tortoise will be a step ahead. No matter how long he runs, Achilles will never reach the tortoise!

Fig. 1 Achilles chasing the tortoise. Achilles begins at $${\color{black}{A_0}}$$ moving to the right with twice the speed of the tortoise, $${\color{black}{v_A = 2v_T}}$$; the tortoise begins at $${\color{black}{T_0}}$$. At later times, whenever Achilles reaches the position that the tortoise held at a previous time, the tortoise has moved a little bit ahead. In infinitely divisible motion, Zeno reasoned that this sequence continues indefinitely and Achilles never catches him.

If we examine the distance between Achilles and the tortoise at each segment, we see that it is decreasing in proportion to the pattern: 1, 1/2, 1/4, 1/8, ..., so that the total distance that Achilles must run to catch the tortoise is in proportion to $${\color{black}{d = 1+ 1/2+ 1/4+ 1/8 + \cdots}}$$. More generally, for any ratio of speeds $${\color{black}{\gamma = v_A/v_T}}$$, we have the sum: $${\color{black}{d = 1+ 1/\gamma + 1/\gamma^2 + 1/\gamma^3 + \cdots}}$$. Zeno's reasoning is essentially thus: if motion is discretized (finitely divisible), then this series terminates because there is a smallest possible step that Achilles can take. With only finitely many steps to take, there is no problem in reaching the tortoise and there is no paradox. If, however, motion is continuous, then there are an infinite number of steps -- each of finite length -- with the result that Achilles must run an infinite distance in order to catch the tortoise. Since, in practice, Achilles succeeds in reaching the tortoise after traversing a finite distance, we apparently have a contradiction. To help visualize this situation, consider the equivalent problem of a square tiled with geometrically smaller rectangles.

Fig. 2 Mathematical problem equivalent to "Achilles and the Tortoise''. The area of the square is finite, as is the distance Achilles physically runs in order to catch the tortoise. However, according to Zeno's reasoning, the sum of the areas of the interior rectangles, just like the total length of Achilles' incremental steps, is infinite.

The above conclusion amounts to the claim that, while the square has a finite area, the sum of the areas of the interior rectangles is infinite. Zeno interpreted this absurdity as evidence that motion must be discontinuous, and this is certainly the only reasonable conclusion given the choices.

But the observation that Achilles must travel an infinite distance is based on the naive assumption that $${\color{black}{d = 1+1/\gamma+1/\gamma^2+1/\gamma^3+\cdots = \infty}}$$, but this is wrong! At least it is wrong when $${\color{black}{\gamma > 1}}$$, which is the condition that Achilles runs faster than the tortoise. I will show a proof of this in a moment, but first let's look at a simple example to make the following remarkable fact clear: the sum of an infinite series of numbers can be finite. For this example, let's assume that $${\color{black}{\gamma = 10}}$$, so that we have the infinite series, \begin{eqnarray*} d &=& 1 + \frac{1}{10} + \frac{1}{100} + \frac{1}{1000} + \cdots \nonumber \\ \nonumber\\ &=& 1 + 0.1 + 0.01 + 0.001 + \cdots \nonumber \\ \nonumber\\ &=& 1.111\cdots \nonumber. \end{eqnarray*} It should be immediately obvious that this sum is not infinite: each successive term only affects the next highest decimal place. After one million terms, we have $${\color{black}{1.11111\cdots}}$$ -- out to the $${\color{black}{999,999^{th}}}$$ decimal place. In the limit that we add together an infinite number of such terms, we arrive at the finite sum $${\color{black}{d=10/9}}$$. As we'll now show, the same thing happens for any series whose terms are geometrically decreasing -- getting smaller and smaller by a fixed factor, $${\color{black}{\gamma}}$$.

Let's first consider a series with only a finite number of terms. The sum, $${\color{black}{d}}$$, is \begin{eqnarray*} d &=& \sum_{i=0}^{n-1} \frac{1}{\gamma^i} \nonumber \\ &=& 1 + \frac{1}{\gamma} + \frac{1}{\gamma^2} + \cdots + \frac{1}{\gamma^{n-1}}. \nonumber \end{eqnarray*} Next, multiply $${\color{black}{d}}$$ by $${\color{black}{1/\gamma}}$$, $$d/\gamma = \frac{1}{\gamma} + \frac{1}{\gamma^2} + \cdots + \frac{1}{\gamma^{n}} \nonumber$$ and subtract it from $${\color{black}{d}}$$: $$d - d/\gamma = 1 - \frac{1}{\gamma^n}, \nonumber$$ so that $$\label{forderiv} d = \frac{\gamma^n - 1}{\gamma^{n-1}(\gamma-1)}.$$ If we then take the limit $${\color{black}{n\rightarrow \infty}}$$, we obtain the sum of the infinite series $$d = \sum_{i=0}^{\infty} \frac{1}{\gamma^i}= 1 + \frac{1}{\gamma} + \frac{1}{\gamma^2} + \cdots + \frac{1}{\gamma^{n}} + \cdots \nonumber = \frac{\gamma}{\gamma -1}. \nonumber$$ This proves the distance $${\color{black}{d}}$$ is finite. For the case of $${\color{black}{\gamma = 2}}$$, we have $${\color{black}{d= 2}}$$, and we confirm that $${\color{black}{d=10/9}}$$ for $${\color{black}{\gamma = 10}}$$, as in our example above. In general, as we increase $${\color{black}{\gamma}}$$, the sum gets smaller (approaching $${\color{black}{d=1}}$$ as $${\color{black}{\gamma \rightarrow \infty}}$$), because the larger the value of $${\color{black}{\gamma}}$$, the faster Achilles runs relative to the tortoise, and so he doesn't need to run as far to catch up.

So Zeno was wrong when he supposed that the infinite series $${\color{black}{1+1/\gamma+1/\gamma^2+1/\gamma^3+\cdots = \infty}}$$. We've just shown that for $${\color{black}{\gamma > 1}}$$, the sum is actually finite, and the series is said to converge 1 . The verdict, then, is that Achilles reaches the tortoise regardless of whether motion is infinitely divisible or not. If it is, while it's true that Achilles takes an infinite number of steps (crosses an infinite number of points $${\color{black}{T_i}}$$), by studying the infinite series, we see that the size of the steps, and hence, the amount of time taken at each one, is geometrically decreasing. The total distance traversed, and the time taken to do it, is finite.

In conclusion, I should point out that a consistent application of Zeno's reasoning throughout the example of "Achilles and the Tortoise'' reveals that, not only does Achilles never catch the tortoise, he never moves beyond his starting point, $${\color{black}{A_0}}$$! Achilles' ability to reach any point, say $${\color{black}{T_0}}$$, is equally stymied by Zeno's argument as his ability to catch the tortoise. The fact that the tortoise is moving in this scenario makes for a livelier illustration of the problem, but it's really not of vital importance to the structure of the paradox. In order for Achilles to reach $${\color{black}{T_0}}$$, he must first pass the midpoint: $${\color{black}{(T_0 - A_0)/2}}$$.
Fig. 3 Zeno's dichotomy paradox asserts that motion is not possible, because and infinite number of midpoints lie between any two points (here, the points 0 and 100). It is essentially equivalent to the "Achilles and the Tortoise'' scenario, and is likewise resolved by convergent series.

But then there's the midpoint between this midpoint and Achilles' starting position, and so on and so on. Because there are an infinite number of midpoints to cross between Achilles and any point a finite distance away, he must take an infinite number of steps to get anywhere. This came to be known as Zeno's dichotomy paradox, but it's essentially equivalent to "Achilles and the Tortoise'' -- we get the same series just in a different order: $${\color{black}{d = \cdots + 1/16 + 1/8 + 1/4 + 1/2 + 1}}$$. This paradox, too, is therefore resolved.

Further Reading: An excellent, in-depth discussion of Zeno's paradoxes can be found in the Stanford Encyclopedia of Philosophy: http://plato.stanford.edu/entries/paradox-zeno/