Zeno's Paradox of Plurality

  1. A line segment can be divided into infinitely many non-overlapping points.
  2. Each of these points has either zero length or some non-zero finite length.
  3. The length of a line segment equals the sum of the lengths of all non-overlapping parts of which it is composed.
  4. If the lengths of the points is zero, then the sum of the lengths is zero.
  5. If the lengths of the points is some non-zero finite size, then the sum of the lengths is infinite.
  6. So, a line segment has either zero length or infinite length.

The straightforward resolution to this paradox is to say that premise 1 is false. It is true that a segment can be divided into an arbitrarily large number of finite segments, but arbitrarily large does not imply infinite.

Some may dislike this resolution because they think that a line segment can be made to correspond to an interval on the real line, and then every real number in the interval should correspond to a point on the line. Indeed, this correspondence can be made and provides an ordered mapping, but it drops the meaning of length for a point. A point does not have length because length can only be defined in terms of the distance between two points. Therefore, with such a mapping, if premise 2 is true, then we find that premise 3 is false. This is a reasonable way to resolve the paradox, but it is not very intuitive. Hence the standard practice of eliminating infinities before they ever get started.

The only other option is to deny premise 2 by asserting that a point can have infinitessimal length, but this is really just a trick used by mathematicians to address the issue in this paradox, it is not a logical way of solving the problem because infinitessimals don't really exist. Infinitessimals are basically a second mistake that cancels out the original mistake of assuming something is infinitely divisible.

Therfore, we are presented with a choice between denying any of premises 1,2 or 3, depending on which mathematical option we prefer. Modern mathematicians in the field of analysis have replaced references to actual infinities with limits, which corresponds to the choice of denying premise 1.