The ancient Greek philosopher, Zeno of Elea (ca. 490-430 B.C.) is famous for describing a series of paradoxes that challenge the existence of motion. In particular, these paradoxes rely on an assumption that it is impossible to complete an infinite number of tasks in a finite period of time.
The first paradox is known as Achilles and the Tortoise. It describes a situation in which Achilles, a hero for the Trojan war, is in a footrace with a tortoise. Because Achilles is confident, he gives the tortoise a 100 meter head start. Once Achilles commences to run, before he can over take the tortoise, he must first come to the 100m spot which the tortoise occupied when Achilles first started to run. Thus, we say that the first task is for Achilles to run to the 100m mark. Of course, since the tortoise is also running, he is, say, 10 meters beyond the 100m spot. The second task is for Achilles to reach the 110 meter spot. Of course, once he is at this spot, the tortoise has again moved forward, say, 5 meters. This process can be repeated indefinitely, and so Achilles must complete infinitely many tasks prior to catching the tortoise. But, as it is impossible to complete infinitely many tasks in a finite period of time, Achilles can never catch the tortoise.
The second paradox is known as the dichotomy paradox. This is similar to the previous paradox. Suppose that Trish wanted to cross the street. Prior to reaching the other side, she first must arrive half way across the street. This is the first task. Prior to reaching the halfway point, she must reach the quarter way point, the second task. Prior to this, she must reach the eighth way point, and so on. Hence, Trish has infinitely many tasks to complete in order to cross the street, which is impossible. Zeno concludes that this situation arises for any moving body and therefore, motion is impossible.
The final paradox, the arrow paradox, involves dividing up time into infinitely many increments unlike the last two, which divided space. Zeno states that for motion to occur, an object must change the position that it occupies. Imagine an arrow in flight. For any instant in time, the arrow is motionless. That is, the arrow is not moving towards the space it currently occupies, because it is already there. The arrow is also not moving towards any space that it doesn't currently occupy, because for this to occur, time must elapse. However, no time elapses in an instant. Thus, at every instant in time, the arrow is motionless. Because time is comprised entirely of instants, motion must be impossible.
Of course, there have been many attempts are solving these paradoxes. For example, mathematician Hermann Weyl proposed that the assumption that space or time is infinitely divisible is incorrect. That is, there are only a finite number of distances between any two points in space or time. The notion of Planck length or Planck time in physics places a limit on the smallest possible length of space or time. This supports Weyl's proposition, however, the Planck length and time have not, to date, been experimentally verified and so, while theoretically significant, the physical significance, if any, remains to be shown. Despite many such attempts at untying Zeno's paradoxes, some modern philosophers still believe that they present relevant metaphysical problems worthy of discussion.