What is Achilles and the Old Turtle (Zenon's Paradox)?

Zenon, one of the ancient Greek philosophers, was a man with interesting ideas who lived between 495 BC - 430 BC. One of them was the idea that all kinds of movements in nature are actually impossible. He tried to prove this idea with a kind of thought experiment involving a running race between Achilles, the fastest runner in Greece, and an old tortoise. Let us now go into the details of this thought experiment;

Since Achilles is the fastest runner of his time, he even gives the old tortoise a 100-metre head start before the race starts. After the race starts, Achilles closes this 100-metre gap in, say, 10 seconds and reaches the point where the turtle started the race. Considering that the men's 100 metres world record in athletics belongs to Jamaican athlete Usain Bolt with 9.58 seconds, we can say that Achilles was also a very fast runner. Of course, the tortoise will not stop at the point where it started the race within this 10-second period and wait for Achilles to catch up with it, it will advance a little bit within 10 seconds, although not as much as Achilles. Let's say that the tortoise advances 10 metres in this 10-second period. This is quite a high speed for a tortoise, but for the sake of ease of calculation, let's assume that this is the case for now. In the light of these assumptions we have made, we can say that Achilles' speed is 10 metres per second and the speed of the tortoise is 1 metre per second. In order to visualise the race better, let us carefully examine the diagrams below;

10 seconds after the start of the race, the tortoise is 10 metres ahead of Achilles and 110 metres from the starting point. When Achilles closes this 10-metre difference in 1 second and reaches the 110th metre, the tortoise will not stay in place in this 1 second, but will move 1 metre further, so it will be 111 metres away from the starting point and will still be 1 metre ahead of Achilles, as shown in the last diagram above.

Based on this logic, Zenon concludes that whenever Achilles reaches the point where the tortoise is, the tortoise will always be ahead of Achilles, since it will not stay in place but will advance a little bit. Since this situation will go on like this forever, Achilles will never be able to catch the old tortoise and never overtake it. Zenon expressed this thought in the following words; ‘Friends, if even the fastest man alive does not have the chance to overtake an old turtle, then the concept of motion is completely absurd.’

However, we all know that this is not the case in real life. For example, even a child, even if he walks, let alone runs, can catch and pass an old turtle very easily under these conditions. How can we solve this contradiction, known as Zenon's paradox? Where is the error in this strange result, which Zenon found by reasoning in the thought experiment described above, but which contradicts real life?

If you wish, let's go back to Zenon's thought experiment and take a closer look at the event step by step. In the first stage, 10 seconds elapsed and when Achilles advanced 100 metres and reached the 100th metre, the tortoise also advanced 10 metres and reached the 110th metre. In the second stage, the elapsed time is 1 second. During this time, when Achilles advances another 10 metres and reaches the 110th metre, the old tortoise advances 1 metre and reaches the 111th metre. In the third stage, the elapsed time is 1/10 of a second. Achilles advances 1 metre in this time and reaches the 111th metre, while the old turtle advances 1/10 metre = 0.1 metre in this time and reaches the 111.1th metre.

According to Zenon, these stages will go on forever and Achilles will never overtake the tortoise, even if the difference decreases each time. According to Zenon, the number of stages is infinite, which means that Achilles will need infinite time to catch the tortoise. But does the infinite number of stages really imply that time is also infinite?

Since the duration of the 1st stage is 10 seconds, the duration of the 2nd stage is 1 second, the duration of the 3rd stage is 1/10 second, we can generalise and express the duration of any nth stage (let us denote it by tn) as follows;

You can test the correctness of this equation by substituting the numbers 1, 2 and 3, respectively, for n, which indicates the number of steps.  I would like to point out that the equation 1/an = a-n, which is valid for a non-zero number a, and the equation an am = an+m are used to express our equation. Accordingly, if we denote the total time from the beginning to the end of any nth stage by Tn , we can write the following equation;

Because Tn is equal to the sum of the durations of all stages from 1 to n. If we substitute the expression we found above for tn in this equation, we can also write our final equation as follows;

When the number of stages n approaches infinity, the number of terms to be summed in our equation above will also approach infinity. Will Tn (i.e. total time), which is the result of this infinite sum, be infinite or finite as predicted by Achilles? If it is finite, what will be its value? Let us now try to find the answers to these questions.

If, as n approaches infinity, the value of Tn also approaches infinity, then Zenon will be right and Achilles will be able to catch the tortoise only after an infinite time. This is equivalent to saying that Achilles will never catch the old tortoise. But if the value of Tn approaches a certain finite number as n approaches infinity, then Achilles will be able to catch and overtake the old tortoise at the end of this finite time. In this case, the contradiction will already be solved.

When we divide both sides of the equation by the number 10 in the last equation, we obtain the following result;

Here, provided that a ≠ 0, the equations (1/a)n = a-n and anam = an+m are used. Note also that the power of the penultimate term is -(n-2). 

Now let us try to calculate the difference between Tn and (Tn /10). Our aim here is to find another expression for Tn. If we use the expressions we found before for Tn and Tn/10 and eliminate the terms that cancel each other, we get the following result;

If the term on the left side of the equation is bracketed Tn, we obtain the following equation;

From here we can find Tn as follows;

This equation will save us a lot of addition to calculate Tn. What we want to find now is what the value of Tn will be as n approaches infinity in this equation (i.e. when n →∞).  If we want to express this in mathematical symbols;

The leftmost term in the above equation indicates the limit value that Tn will approach as n approaches infinity. To calculate this value, we need to look at what the value of 10-(n-1) in the denominator in the above equation will be as n approaches infinity. In other words, we need to find an answer to the following question;

When n is a very large number approaching infinity, the value of n-1 and therefore the number 10(n-1) will also approach a very large value. Therefore, the number 10-(n-1) = 1 / 10(n-1) will also be a very, very small number approaching zero. Then we get the following result;

If we substitute this into our equation, we get the following result;                                 

Accordingly, even if the number of stages n is infinite, the total time, Tn, is a finite number and 11,11111. . . . seconds later Achilles will catch the old tortoise and then (for example at the 12th second) he will have overtaken him. Zenon was mistaken in thinking that if n is infinite, then Tn will also be infinite. Therefore, there is actually no contradiction. From this point of view, we can also calculate at what metre Achilles can catch the old tortoise. Since Achilles is moving at a speed of 10 metres per second, 11,11111. . . . in exactly 111,11111. . . . . . metres and will catch the old turtle at exactly 111,11111. . . . . . . . th metre.

This example is very important as it is an interesting example of how some logical errors can lead us to wrong conclusions and create contradictions.