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# Zeno's Paradox Achilles and tortoise

An eternal competition between Achilles and a tortoise.

Part 4. Refuter: Achilles does not go through an infinite number of points one by one

Part 5. Paradox Defender: Paradox and theory of relativity or can we find conditions that paradox still apply?

Conclusion

Part1.

One of the most famous Zeno’s paradox is a paradox about a perpetual race between Achilles, the best runner in Greece, and an old lazy tortoise. This paradox has been solved, discussed, explained, declined during more then twenty four centuries, time enough to congratulate it (paradox) with an immortality, or at least accept it as a long-lived tale.

The sense of the paradox is:
Achilles, a symbol of quickness must overtake a tortoise, symbol of slowness. Achilles runs ten times as quick as the tortoise and gives her ten metros odds.
Achilles runs ten meters, a tortoise runs one meter,
Achilles runs one meter, a tortoise runs one decimeter,
Achilles runs one decimeter, a tortoise runs one centimeter.
And so on until the infinity. As a result according to Zeno Achilles can run everlastingly, but never will be able to run down a tortoise.
Starting from Aristotle until 20 century there were a lot of attempts to refute this paradox, but no one refutation seemed ideal.
Is this paradox really irrefutable? In reality Achilles overtakes a tortoise very easily and even does not suspect that the famous paradox forbids him to fulfil such an illegal action.

Let us consider point of view of refuter of the paradox and paradox defender.

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Part 2. Refuter:

Point of view of refuter:
The mystery of the paradox is in its haze formulation. Zeno in a disguised way gives us a limitation on the length and time of the race.
Let us be fair to Zeno. Soon of all he did not intend to delude descendants with a help of haze formulation. Perhaps he did not realize it himself.
To solve this paradox it is necessary to consider an infinite geometric series and know that a sum of the infinite series could be a finite digit. Necessary actuarial mathematics was introduced by Newton and Leibniz many centuries later. In the times of Zeno philosophers guessed that the sum of infinite series was infinite too.
Zeno offered us to consider a series:

and so on until the infiniteness.

This sequence is a positive convergent geometric series.

Scale factor and common ratio :

A sum S is equal to:

In our case:

It means Achilles runs over the tortoise in 100/9 meters from his first position and as his speed is 10 m/sec it happens in 10/9 seconds.

With the given terms every new interval: centimeter, millimeter, micron and so on increases the length of the race by less and less value and every new partial sum is less than its integral sum of 100/9 meters. It will be so until time 10/9 seconds expires.
Such way Zeno had given us the length of the race, limited 100/9 meters and time of the race 10/9 seconds.

Refuter's conclusion:
Zeno’s Paradox stops being a paradox if honestly to add a couple of words to its formulating: Achilles is not able to overtake a lazy tortoise if the length of the race is less than 100/9 meters or the time of the race is less than 10/9 seconds.
If Achilles isn't given time to run he will not move at all.

An internal core of the paradox

Point of view of refuter:
Calculation of the moment of "meeting" of Achilles and tortoise is not a subject of the paradox. What is at the core of Zeno's paradox is the idea that in order for Achilles to catch up with the Tortoise, he must first perform an infinite number of acts. And, that seems to be impossible.

Really, it is not obligatory to consider infinite series to get the answer of 100/9 meters. It is possible to solve a problem algebraically.
Let x is a distance which was crept by Tortoise till the moment of a meeting with Achilles.
Then 10 + x is a distance which was run by Achilles till the moment of a meeting.
Speed of Achilles is 10 m/sec, speed of Tortoise is 1 m/sec.
Time of movement from start up to a meeting is the same for both runners.
Let us write an equation.
x/1 = (10 + x)/10.
x = 10/9 is the distance which was crept by Tortoise till the moment of a meeting.
10 + x = 100/9 is a distance which was run by Achilles till the moment of a meeting.

It is difficult to believe, that Zeno could not find a solution. It is even more difficult to imagine, that Zeno never overtook nobody or did not see, how it is done by others.

Calculus allows us to calculate the sum of an infinite numbers of terms, but does not explain how somebody is capable to finish going through an infinite number of points, if he has to go through these points one by one.

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Part 4. Refuter:
Achilles does not go through an infinite number of points one by one

Refuter's argument:
Achilles does not need to go through an infinite number of points one by one.
The first section is 10 meters and Achilles has to run 10 - 20 steps to get over it.
The second section is 1 meter and Achilles has to run couple steps to get over it.
The third section is only 1 decimeter. That's it. Achilles's foot is bigger then third, fourth, fifth and so on sections and Achilles covers all these sections simultaneously with one step. He does not go through an infinite number of sections one by one.

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Paradox and theory of relativity or can we find conditions that paradox still apply?

Well, Achilles does not go through an infinite number of points one by one if he runs with terrestrial speed. But we may make him to do it if his speed is commensurable with a speed of light.

Let us try to find such conditions, that paradox still apply. To do it we will have to consider the famous paradox from the point of view of theory of relativity.

First of all, we have to deprive Achilles of the ability to cover some space sections simultaneously with one step, and to make him go through an infinite number of sections one by one.
How to do it?
We have to find a way to decrease Achilles's size with each new smaller section. Such way he will have to run each new section despite the length of this section is very small.

In accordance with Einstein’s theory of relativity the length of any object in a moving system will appear foreshortened in the direction of motion. The higher speed the less size. The length has its maximum in the system in which the object (Achilles in our case) is at rest. If L0 is a length of Achilles in this system then the length L in the moving system is
,
where c is speed of light, V is speed of Achilles.
If speed V of Achilles is commensurable with a speed of light then length’s decreasing will be substantial.
Of course, even the quickest Achilles is not able to run so quickly. Well, let place our famous runner in a rocket.

Speed of Achilles’s rocket is commensurable with a speed of light.
L = 0.1L0 when V = 0.95c

To keep decreasing, Achilles' s rocket has to fly with acceleration.
L = 0.1L0 when V = 0.95c

L
= 0.01L0 when V = 0.995c
L
= 0.001L0 when V = 0.9995c
L
= 0.0001L0 when V = 0.99995c
And so on.

What will be with time?
A clock in a moving system will be seen to be running slow. The time will always be shortest as measured in its rest system. The time measured in the system in which the clock is at rest is called the "proper time".
,
where T0 is "proper time" interval.
T = 4.47T0 when V = 0.95c

T
= 14.14T0 when V = 0.995c
T
= 44.72T0 when V = 0.9995c
T
= 141.42T0 when V = 0.99995c
And so on.

Events from point of view of detached observer:
Seeming decreasing of length in the direction of motion and slowing down of time will be from the point of view of detached observer.
Destination between Achilles and tortoise decreases. Achilles appears thinner and thinner, tortoise is closer and closer, but now foreshortened rocket is not able to cover some space sections simultaneously, and Achilles in his rocket has to fly through an infinite number of points one by one. It makes Zeno right.

Events from point of view of Achilles:
Achilles himself will not feel any changes inside a rocket, he will not feel that rocket moves. For him the rocket itself and tortoise ahead will be "frozen", and he will see that all surroundings rush back.

What happened with Achilles relativistic mass?
It will increase.
The increase of effective relativistic mass is given by the expression
,
where m0 is "rest mass".

So, the length of Achilles’s rocket continues to decrease in the direction of motion. Relativistic mass continue to increase, the time is slow down. It is a state of singularity.

Paradox Defender: Aim is achieved. Achilles can not overtake a tortoise.

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Conclusion

Calculation of the moment of "meeting" of Achilles and tortoise is not a subject of the paradox.

A real core is a question is an eternal division of the length of section possible? In other words is space discrete or continuous?

When we calculate the moment of "meeting" of Achilles and tortoise we do not answer real Zeno's question, we simply disregard it.

We can ignore true sense of paradox and neglect a question about "discrete or continuous space" while we consider terrestrial speeds. Even the fastest Achilles is not an exception.

To make Zeno right we made Achilles in the rocket to “jump” into singularity state. Now Achilles in his rocket has to go through an infinite number of sections one by one.
Now the question of "continues or discrete space" is important. If space is continuous then rocket with Achilles will run eternally.
If space is discrete then … what will happen when Achilles’ rocket achieve the smallest last Rubicon?

There is the third possibility. We will talk about it later.

by Tetyana Butler

Bibliography

[1] Huggett, N. Zeno's Paradoxes; The Stanford Encyclopedia of Philosophy (Summer 2004 Edition), Edward N. Zalta (ed.); http://plato.stanford.edu/archives/sum2004/entries/paradox-zeno/

[2] Aristotle Physics; Book 8; Translated by R. P. Hardie and R. K. Gaye; http://www.abu.nb.ca/Courses/GrPhil/Physics.htm

[3] Brown, K., Reflections on Relativity; http://www.mathpages.com/rr/rrtoc.htm

[4] Saari, D. G. and Xia, Z. Off to infinity in finite time; AMS Notices 1995

[5] Isham, C. J. Canonical quantum gravity and the problem of time; http://arxiv.org/abs/gr-qc/9210011

[6] Unruh, W. G. Time, gravity, and quantum mechanics; http://arxiv.org/abs/gr-qc/9312027

[7] Wang, Z. Y. and Chen, B. Time in quantum mechanics and quantum field theory; J. Phys. A 2003

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