Before an object can travel a distance d,it must keep traveling “in halves”: In terms of
the racetrack, in order to reach the end of the course, a person would have to first reach
the halfway mark, then the halfway mark of the remaining half, then the halfway mark
of the final fourth, then of the final eighth, and so on ad infinitum(to infinity). There-
fore, the distance can never truly be traveled to reach the end of the racetrack.
The Achilles and the tortoise paradoxis a version of the tortoise and the hare, but
with a very different resolution than the well-known fable. In this paradox, Achilles
gives the slower tortoise a head start; Achilles starts when the tortoise reaches point a.
But by the time Achilles reaches a, the tortoise has already moved beyond that point,
to point b; when Achilles reaches b, the tortoise is at point c, and so on ad infinitum.
Since this process goes on forever, Achilles can never catch up with the tortoise.
Another paradox is the arrow paradox. In this case, an arrow in flight has a cer-
tain position at a given instant in time, but that is indistinguishable from a motionless
arrow in the same position. So how is the arrow’s motion perceived?
Finally, one of the most interesting and insightful paradoxes is attributed to
Socrates—thus, it is called the Socrates’ paradox. It is based on Socrates’ statement,
“One thing I know is that I know nothing.”
121
FOUNDATIONS OF MATHEMATICS
When a horse and jockey race around a track, as they circle they must repeatedly cut the distance to the finish
line in half. The dichotomy paradox says that if this is literally the case, the horse will never complete its race; it
will just make the distance to the finish line smaller and smaller. Taxi/Getty Images.