In Achilles and the Tortoise, when Achilles pursue the tortoise, he is not "counting" the point in space.
In the linear space continuum, points are not "isolated" entities : they are cuts. We have a "place before" (i.e. all numbers before SQRT(2)) the cut, and a "place after" the cut (all numbers after SQRT(2)) [this is Dedekind' analysis of the continuum : see The Continuum and the Infinitesimal in the 19th Century].
Achilles will overtake the tortoise simply running (for example) twice faster than the tortoise: we do not need to assume that he has to run faster and faster ("count the natural numbers by counting faster and faster and faster").
We need to use a device to measure the progress of the run: if we use the heartbeat, and assume that the tortoise starts an heartbeat before Achilles, after the first interval in time dt, she [he, it ?] will have traversed a certain amount of space ds. With the second heartbeat Achilles will start, and we assume that he runs twice as faster as the tortoise. After the second heartbeat (i.e.after 2 x dt), both Achilles and the tortoise will have traversed a space equal to 2 x ds. After the third heartbeat, Achilles will have definitively outrun the tortoise (he have traversed a 4 x ds, while the tortoise has only traversed 3 x ds): this has required a finite amount of time (3 x dt) and without the need of an "unlimited increasing" speed.
If we model the race with the mathematical continuum we must not make the mistake of describing the progress of the runner as made of successive move from on point to "the next": in the real number line, a point has no "next".
We may say that Achilles will win because he is not counting the point in space; he is "traversing" intervals in time.
My personal "feeling" with this paradox is the same as the proposed solution in Wiki Zeno's paradoxes :
Pat Corvini offers a solution to the paradox of Achilles and the tortoise by first distinguishing the physical world from the abstract mathematics used to describe it. She claims the paradox arises from a subtle but fatal switch between the physical and abstract. Zeno's syllogism is as follows:
P1: Achilles must first traverse an infinite number of divisions in order to reach the tortoise;
P2: it is impossible for Achilles to traverse an infinite number of divisions;
C: therefore, Achilles can never surpass the tortoise.
Corvini shows that P1 is a mathematical abstraction which cannot be applied directly to P2 which is a statement regarding the physical world. The physical world requires a resolution amount used to distinguish distance while mathematics can use any resolution.
See also Zeno's Paradox for further refences and Wesley Salmon, Zeno's Paradoxes, (2nd Ed - 2001).
