Zeno's Paradox

Zeno's paradoxes are ancient paradoxes in mathematics and physics. Using seemingly analytical arguments, Zeno's paradoxes aim to argue against common-sense conclusions such as "More than one thing exists" or "Motion is possible." Many of these paradoxes involve the infinite and utilize proof by contradiction to dispute, or contradict, these common-sense conclusions.

In many cases these paradoxes went unresolved for centuries, or have solutions that are still debated. In general, the epsilon-delta definition of a limit proves that infinite series can produce finite results, and serves as a baseline for disproving these paradoxes. The problem that follows is known as Thompson's lamp, developed in 1954, and is evidence to the longevity of these paradoxes. Zeno's paradoxes have remained part of the mathematical conversation for over 2,400 years.

Motion is impossible. \(_\square\)

Progression Version:

Suppose that the path from A to B is of an unit length.
To travel from A to B, he must cover half the distance first.
After he has covered half the distance, he must still cover half of the remaining distance, that is a quarter.
And then, he must also cover half of that distance and ad infinitum.
Hence, motion is impossible because of the following plausible reasons:

• There are an infinite number of tasks.
• There is no definite last step. \(_\square\)

Regression Version:

To complete the travel from A to B, he must cover a unit distance.
But before that, he must cover half the distance first and even before that half that distance.
Hence, motion must be impossible because

• that would be an infinite number of tasks;
• there is no definite first step. \(_\square\)

Motion is impossible. \(_\square\)

Suppose an arrow is shot which travels from A to B.
Consider any instant.
Since no time elapses during the instant, the arrow does not move during the motion.
But the entire time of flight consists of instances alone.
Hence, the arrow must not have moved. \(_\square\)

The fastest runner of antiquity, Achilles, would not be able to overtake the tortoise in a race when the tortoise has a head start. \(_\square\)

Construct the \(x\)-axis along the race track.

Suppose that at the beginning Achilles is at \(x_0\) and the tortoise is at \(x_1\) such that \(x_0\) is to the left of \(x_1.\)
For all \(n \geq 1\), when Achilles is at \(x_n\), the tortoise is at \(x_{n+1}\), where \(x_n\) is to the left of \(x_{n+1}\), and Achilles does not overtake the tortoise at any point between \(x_{n-1}\) and \(x_n\).
Hence, Achilles cannot overtake the tortoise because

• that would take too long;
• that would be an infinite number of tasks;
• there is no \(n\) where \(x_n\) is to the left of \(x_{n-1}.\) \(_\square\)

See Infinity for a rigorous treatment.

Before exploring Zeno's paradoxes of motion in further detail, we need to notice that the fundamental problem arises from the problem of infinite division versus infinite counting.

To understand what is going on, we categorize infinite sets into two types:

• countably infinite: an infinite set which can be enumerated
• uncountably infinite: an infinite set which cannot be enumerated.

It can be rigorously shown that continuous or infinitely divisible sets are actually unncountably infinite, which is the case with space and time (at least in classical physics or as relevant to this problem). In fact the continuum is at the heart of real analysis.

One of the primary fallacies of Zeno's argument is that he essentially tries to break down space and time into discrete components and thereby enumerate them, which is what cannot be done with such continuous objects.

Hence, asking which is the first step is not really a valid question as long as space and time are continuous.

Note that it is possible to do an uncountably infinite number of tasks because time, like space, is continuous as well.

See Convergence of Sequence for rigorous treatment.

Zeno often claims that motion can never be accomplished because Achilles has to move an infinite number of positive steps.

Though the number of steps are infinite, their sum is still bounded by a finite number, which is what we call convergence of series.

For any \(n\), \(\displaystyle \sum_{i=1}^{n} \left(\frac{1}{2}\right)^i\) is equal to \(1-\left(\frac{1}{2}\right)^n,\) which tells us that the series converges to \(1.\)

See Epsilon-Delta Definition of Limits for a rigorous treatment.

As you might have expected, the arrow paradox is a division by zero problem.

First, Zeno defines something to be in motion when the quantity \(v = \frac{\Delta s}{\Delta t} \) is positive. Then, he asserts that \({\Delta s}\) is zero at an instant, and claims \(v\) to be zero, which is absurd.

The solution is to replace velocity with \[ v = \lim_{\Delta t \to 0} \frac{\Delta s}{\Delta t}, \] which means that \(v\) is the value which the average velocity approaches as the concerned time duration shrinks towards zero.

As Russel proposed, there is no motion at a duration-less duration. Instead, the motion must be described in terms of their velocities. Note that our new definition of velocity does allow us to define velocity at an instant.

Half a given duration is equal to double that duration. \(_\square\)

THE STD PARADOX

Suppose that A, B, and C are bodies of equal size stacked one below another as in the figure.
While the row of A is stationary, those of B and C are moving to the right and left, respectively, with the same speed.
By the time the rightmost B crosses 1 A, it actually crosses 2 C's.
Since A and C are of equal length, half a given duration is equal to double that duration. \(_\square\)

The fallacy here is that a body in motion does not pass things in rest and things in motion at the same velocity.

This is the key to the concept of relative velocity. If the speeds of B and C were \(v\), then the relative speed of one with respect to the other would be \(2v\), thus resolving the paradox.