As requested by jreelawg for the original OP dainamiku here is an explanation of two Zeno's paradoxes. It is not pssible for me to praphrase the text to just deal with Achilles and the tortoise. Bear in mind that Reichenbach, as far as i know, is a philosopher rather than a physicist.
I have copied this extract in full although it may not all be relevant.
From Hans Reichenbach. The Direction of Time. Pages 5-6. Published in 1956 but this edition in 1971.
Zeno’s paradoxes of motion have often been discussed. He argues that if motion is travel from one point to another, a flying arrow cannot move as long as it is at exactly one point. But how then does it get to the next point? Does it jump through a timeless interval? Obviously not. Therefore motion is impossible. Or consider a race between Achilles and a tortoise, in which the tortoise is given a head strart. First Achilles has to reach the point where the tortoise started; but by then, the tortoise has moved to a farther point. Then Achilles has to reach that other point, by which time the tortoise again has reached a farther point; and so on, ad infinitum.Achilles would have to traverse an infinite number of non zero distances before he could catch up with the tortoise; this he cannot do, and therefore he cannot overtake the tortoise.
Concerning the arrow paradox, we answer today that the rest at one point and motion at one point can be distinguished. “Motion” is defined, more precisely speaking, as “travel from one point to another in a finite nonvanishing stretch of time”; likewise, “rest” is defined as “absence of travel from one point to another in a finite nonvanishing stretch of time”. The term “rest at one point at one moment” is not defined by the preceding definitions. In order to define it, we define “velocity” by a limiting process of the kind used for a differential quotient; then “rest at one point” is defined as the value zero of the velocity. This logical procedure leads to the conclusion that the flying arrow, at each point, possesses a velocity greater than zero and therefore is not at rest. Furthermore it is not permissible to ask how the arrow can get to the next point., because in a continuum there is no next point. Whereas for evry integer there exists a next integer, it is different with a continuum of points; between any two points there is another point. Concerning the other paradox, we argue that Achilles can catch up with the tortoise because an infinite number of nonvanishing distances converging to zero can have a finite sum and can be traversed in a finite time.
These answers, in order to be given in all detail, require a theory of infinity and of limiting processes which was not elaborated until the nineteenth century. In the history of logic and mathematics, therefore, Zeno’s paradoxes occupy an important place; they have drawn attention to the fact that the logical theory of the ordered totality of points on a line—the continuum—cannot be given unless the assumption of certain simple regularities displayed by the series of integers is abandoned. In the course of such investigations, mathematicians have discovered that the concept of infinity is capable of a logically consistent treatment, that the infinity of points on a line differs from that of the integers, and that Zeno’s paradoxes are not restricted to temporal flow, since they can likewise be formulated and solved for a purely spatial continuum.
As a footnote he adds. For a modern treatment of Zeno’s paradoxes , See Bertrand Russel, Our Knowledge of the External World.
Matheinste
