## Foundations of Mathematics## Axiomatic System |

## What are some paradoxes that deal with space and time? |

There are numerous paradoxes that deal with the counterintuitive aspects of continuous space and time. One of the most well-known is the *dichotomy (or racetrack) paradox.* 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). Therefore, the distance can never truly be traveled to reach the end of the racetrack.

The *Achilles and the tortoise paradox* is 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 certain 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, the *Socrates’ paradox.* It is based on Socrates’ statement, “One thing I know is that I know nothing.”