Backwards Zeno's Paradox

Question

As I understand it, correct me if I am wrong, walking a finite distance will not take an infinite amount of time, because although you have to travel an infinite number of finite distances, the time it takes you to travel those distances is proportional to them, and so your overall time will not be infinite. But could you also say that traveling any distance takes no time at all? If you walk along a line, which is composed of infinitely many points, the time to travel a single point is 0, so even if you travel infinitely many of them your time is still 0. I must be wrong, I'm just not sure how.

Answer 1

Here's the solution of the paradox according to measure theory. There are other "alternative" solutions which also work, but this solution is the most popular one.

The basic solution is: The whole can be greater than the sum of its parts!

Or, more precisely: if I have an interval or something $S$, and I divide $S$ into a bunch of pieces, then the length of $S$ may be greater than the sum of the lengths of those pieces.

Usually, we think of properties like weight and length as being additive. If I put a bunch of objects on a scale, then the weight of the entire collection of objects is equal to the sum of the weights of the individual objects. If I make a journey composed of segments, then the length of the journey is the sum of the lengths of all of the segments. Right?

Measure theory says: yes, but only if the number of pieces is countable. In other words, additivity holds if there are finitely many pieces, or if there are a "small infinite number" of pieces, but it fails if there are a "large infinite number" of pieces.

That's exactly what happens here. Yes, a line segment of length $1$ is composed of infinitely many pieces of length $0$. But it's composed of a "large infinite number" of such pieces.

(There are alternative solutions, too. If I remember right, one alternative solution is to state that a line cannot be divided entirely into pieces of length $0$. But this solution turns out to have other problems, and for that reason, it's less popular.)

Answer 2

The time to travel a single point is a meaningless concept since a point has not dimension.

Given a length $L$ divided in $N$ equal intervals $\Delta L=L/N$ the time to travel each interval at speed $v$ is

• $t_N=\frac{\Delta L}{v}=\frac{L}{vN}$

and of course as $N$ gets larger $t_N\to 0$ but the overall time to travel $L$ at speed $v$ is equal to

• $t=N\cdot t_N=N\cot \frac{\Delta L}{v}=N\cdot \frac{L}{vN}=\frac L v$