A PHILOSOPHY OF THE FOUR-DIMENSIONAL SPACE-TIME
The Worldview of Relative Simultaneity (MURAYAMA Akira)
CHAPTER II The Problems of Continuity and Contradiction
3. Changes in the Distance, the Dichotomy Paradox and Arrow Paradox
Is this true? No, this is not actually true. Rather, this is where the problem begins. With a chuckle on his face, Zeno would say the following.
All right. I understand that you note the distance between Achilles and the tortoise, but just think about this: Before the distance between Achilles and the tortoise becomes zero, the distance has to become half. By the same token, before the distance becomes half, it has to become half of the half. This is an endless cycle. I mean that to shorten the distance between them, they have to pass through an infinite distance. Is this possible within a limited period of time? In the end, Achilles cannot get near the tortoise at all, let alone catch up.
This is Zeno's first paradox, which is the dichotomy paradox. That is, the paradox about Achilles and the Tortoise is inevitably linked to the dichotomy paradox even with the introduction of changes in the distance between Achilles and the tortoise.
Zeno might also say the following: Well, shrinking distance is really not shrinking. This is because shrinking distance has a certain value of length at each moment. At the moment a distance has a certain value of length; it is not contracting. However, the time from a certain state to another state of distance is comprised of each moment during the period. Therefore, shrinking distance is really not shrinking.
This reasoning leads to Zeno's third paradox, the arrow paradox, if "shrinking distance" is changed to an "arrow." Zeno's arrow paradox argues as follows: A flying arrow is really not flying because the flying arrow occupies a certain position at each moment. What occupies a certain position is not flying at the moment, but the time from a certain state to another state of an arrow is comprised of each moment during the period. Therefore, a flying arrow is really not flying. That is, this boils down to the fact that Zeno's paradox about Achilles and the Tortoise leads to his arrow paradox.
Considering in this way, I wonder if I should examine what infinity is and what continuity is, as many others have previously explored.
The concept of limit postulated by Isaac Newton (1642-1727) and Gottfried Wilhelm von Leibniz (1646-1716), the concept of infinite cardinality espoused by Georg Ferdinand Ludwig Philipp Cantor (1845-1918) and the concept of Julius Wilhelm Richard Dedekind's cut (1831-1916) can be described as a series of historical struggles with 0, an inscrutable and awkward number. The modern ages saw the development of many logical and mathematical efforts to incorporate the infinitesimal into a consistent axiomatic system.
The theory construction brought about enormous benefits for humankind. In a sense, Zeno's paradoxes might work as a significant catalyst for boosting these mathematical developments. However, I am skeptical of reducing the solution of Zeno's paradoxes to the subjects of infinity and continuity. This is because infinity and continuity are not peculiar to the subjects of movement and time. It seems that Zeno did not go as far as to say that space expansion was too full of contradictions to recognize.
If you point out contradictions in continuous space expansion, you will have to deny everything in the world; for example, space does not exist and bent things do not exist. Or space can be regarded as an aggregation of discrete existences. This is a way of thinking based on atomism postulated by Dēmokritos (470-370 B.C.). Space expansion can be established even if it is not based on the assumption of continuity, but if you discuss space movements and changes, the subject of continuity matters. This is because you have to consider infinitely intermediate processes of transition from a certain state to another state of space. The core essence of Zeno's paradoxes is constituted by the infinite dichotomy in the processes. In this sense, the subjects of continuity and infinity are tightly linked to movement and changes.
Continuity and infinity cannot be measured and confirmed in principle. If they can be measured to confirm, they would not be infinite. They are regarded as just theoretically thinkable objects. That is, they are concepts that were logically established because the limiting point could not be assumed. I suspect that it was this point that Zeno intended to highlight. Probably, he thought that infinite state could not be realized.