*Originally posted by wesmorris *

**But doesn't calculus in essence state that it doesn't matter, we know he has crossed the room? In essence aren't you asking the ridiculus question "what is infinity minus one?". **

We always knew he crossed the room. Even Zeno knew that. That's why it's a paradox. There are some similar problems that calculus solved. For example, the problem of infinite divisibility.

Calculus shows us how a finite distance can be conceived of as the sum of an infinite sequence of ever decreasing intervals. An infinite number of intervals can sum to a finite distance. In this way Calculus actually shows that Zeno's description isn't the problem. If Achilles traverses each of the infinite number of ever decreasing intervals, he will have done so in a precisely definable finite distance.

But notice that all we get here is a conditional. We still need to show that Achilles *n* traverse each of an infinite number of ever decreasing intevals. That's where the problem is. None of the intervals are a final interval. So there is no interval such that Achilles crosses the road (or catches the tortoise) upon completing that interval. So long as Achilles has just finished an interval, he has not yet crossed the road. But notice that all Achiles does is complete intervals one at a time. So there's nothing he does that can count as completing his crossing of the road.

Imagine you have a bag with an infinite nuber of beans in it, and you have to empty the bag one bean at a time. Fortunately you are able to remove each bean in half the time it took you to remove the previous one. If it takes you 30 seconds to remove the first bean, we can calculate that you'll be all done in 1 minute. (I'm sweeping some issues about cardinality and transfinite subtraction under the rug here.) But how do you finish? You never take the last bean out of the bag. There is no last bean! No matter how many beans you remove, there are always the same number left. So you take out beans faster and faster, never getting any closer to the bottom of the bag, until suddenly, without taking out the last bean, the bag is empty! How is that possible?

If Achilles crosses the road, then there must be something that he does in virtue of which he crosses the road. But nothing he does counts as crossing the road. So he can't doesn't cross the road. But of course we know he does cross the road. Ergo, paradox.