As
whizkid says, the problem with
Achilles and the tortoise is not really one of
logic or
maths, but one of wrong premises or assumptions.
What is the wrong assumption?
What seems to be the problem is that each time Achilles reaches the place where the tortoise started this time-step, the tortoise has moved on a smaller distance.
t0 : A---------t-----------
t1 : ----------A----t------
t2 : ---------------A--t---
t1 : ------------------A-t-
The wrong assumption is illustrated by the figure above. Each line represent a different point in time. But the points in time are not linearly distributed. In fact, we look at smaller and smaller time-steps, since both Achilles and the tortoise have constant speeds (but move shorter and shorter distances). Before Achilles reaches the tortoise, the size of the time-step goes towards 0.
This may or may not be obvious when you think about the paradox the first time. If you keep the time-steps constant, Achilles will naturally pass the tortoise (and indeed, the paradox does not occur).