Solving Zeno's Paradox

For those of you unfamiliar with Zeno of Elea, he was a Greek philosopher in the 5th century BC who proposed a series of paradoxes, mostly about motion, and how in the logic of the paradox it was impossible. I'm going to deal with just one of them, Achilles and the tortoise.

Imagine this scene: the Aegean sun is blazing on the white marble courtyards of Athens and Achilles has decided to race a toroise in it. Despite having rather fragile heels he is a sporting man and will thus allow the tortoise a significant head start on him. As the start of the race is signalled, by Socrates perhaps, asking the race organizer what a contest is, Achilles and the tortoise both set off, the tortoise with blazing fire and cracked cobblestones accompanying its supersonic progress ... oh wait, that's not right is it. Achilles runs, but half way to the tortoise's starting point he has a problem. He's traveled some distance but in that time the tortoise has moved away from him. He keeps on running despite this energy-sapping cogitation and ends up once again covering half the distance between where he was before and where the tortoise was before. Now he is starting to get confused. He's only a grunt after all, not a brain surgeon, and can't quite understand why the tortoise is still twice as far as Zenowill allow him to run. Before long his brain overheats and explodes, sending shrapnel into his heel and thus causing his demise, and the tortoise is declared the winner.

In the lucid prose of Aristotle: "In a race, the quickest runner can never overtake the slowest, since the pursuer must first reach the point whence the pursued started, so that the slower must always hold a lead."

So how are we to solve this paradox? You may have seen solutions involving calculus or algorithms courtesy of Cauchy or Leibniz or other brilliant mathematicians but these are solutions of the problem, not solutions of the paradox. They only show that the paradox can be resolved, not what is wrong with it in the first place.

To make my solution more obvious, I will reword the paradox slightly: "Achilles can never pass the tortoise because he must first travel half the distance between them, then half the remaining distance, and so on for ever."
Now to solve it. Firstly we must examine all the time terms in this statement. Zeno says that Achilles can never reach the tortoise, by which I take it to mean that no matter how much time he is given, he can not pass the tortoise. Secondly, assuming that Achilles's speed is constant, the time of each successive "step" is half that of the previous step. The paradox lies in assuming that it takes an infinite amount of time to do an infinite number of steps. The mathematical solutions will tell you this is not the case. If we only allow finite steps, as specified in the paradox, then yes, Achilles really can't cover the distance. After all, finite steps do not furnish him the required time to close the gap. And then even if we allow the number of steps to go to infinity, the infinite sum of xi = t/(2^i) is t, the time taken for Achilles to reach the tortoise but not to get past. To get past he actually needs more time than the paradox has allowed him. To draw level he needs exactly the time allowed him, but this is far from obvious and it took the genius of Archimedes to work it out.

So it is not really a paradox at all. First Zeno says he has all the time in the world, then he implicitly says that he only has less time than it takes to complete the task in the first place. This is a contradiction, not a paradox, but it's really hard to spot.