Paradox: so the turtle wins the race!

Aristotle and Zeno

The great Greek philosopher Aristotle set the rules of logic, the way of reasoning, more than 2000 years ago. Soon, however, it turned out that something is wrong with the picture of the world Aristotle created. It became evident after Zeno of Elea had produced his famous paradoxes, i.e. results of reasoning (using of logic) which conflict with experience in the real world. It was a pretty big scandal, as Aristotle was a highly honored scholar of ancient Greece.

One of the best known paradoxes of Zeno is about Achilles and the turtle:

Since Achilles was noted for his swiftness and the turtle for its slowness, the turtle is given a head start when they race each other. Zeno argues that Achilles first must reach the point where the turtle was initially. By then, the turtle will have moved beyond that point. Now the situation is the same as it was at the start of the race. The turtle has a head start on Achilles. Achilles must again reach the point where the turtle was (when Achilles reached the point where the turtle got his first head start). But by the time he arrived, the turtle has moved on. Now the situation is the same as it was at the start of the race. The turtle has a head start on Achilles. And so on. Achilles can never catch, let alone pass, the turtle, so the turtle wins the race!

Incredible! But it seems to be O.K. as far as the reasoning is concerned. The
Greeks went nuts.
The thing to blame for this tragedy is called infinity. Indeed, when we say ”and
so on”, we mean that one can do the same thing over and over again infinitely many
times! But then (it makes sense, or what!) it will never end!
This is not true.
We can explain this using a notion from Calculus, so-called geometric series:
Let q be a positive number less than 1. Then the infinite sum
1 + q + q2 + q3 + q4 + · · · + qn−1 + . . .
is meaningful, and we can calculate its value using the formula
1 + q + q2 + q3 + · · · + qn−1 + · · · =
1
1 − q
.
The solution of the above paradox is maybe better understandable with specific
distances and speeds given. Let’s say, that the original head start for the turtle is
100 yards, and that Achilles can run 10 times as fast as the turtle, thus
vAchilles = 10vturtle.
In this case, by the time Achilles reaches the turtle’s starting point, the turtle has
moved 10 = 100
10 = 100×0.1 yards from that point. By the time Achilles reaches the
110-yard point of the race, the turtle will be at 111 = 100+100×0.1+100×(0.1)2
yards. When Achilles is at 111 yards, the turtle is at 111.1 = 100 + 100 × 0.1 +
100 × (0.1)2 + 100 × (0.1)3 yards, and so on. As the race progresses, the turtle is
heading for the point at exactly

100 + 100 × 0.1 + 100 × (0.1)2 + 100 × (0.1)3 + · · · + 100 × (0.1)n−1 + · · · =
= 100 × (1 + 0.1 + (0.1)2 + (0.1)3 + · · · + (0.1)n−1 + . . . ) =
= 100
1
1 − 0.1
=
1000
9
= 111.111 . . . yards.

When they both get to that point, Achilles will catch the turtle at 111.111 . . .
yards, pass it in the next instant, and go on to win the race. Hence, there is no
paradox.