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I read this question How can Zeno's dichotomy paradox be disproved using mathematics? .

The first (ie the one on the top) answer uses the fact that $\sum\limits_{n=1}^\infty\frac{1}{2^n}=1$, because we divided the movement in $1/2+1/4+1/8+...+1/2^n$

However, we could have used another way to decompose the movement: for instance, $1/4+1/9+1/16+...+1/n^2$ etc. However, this sum converges towards $(\pi^2-6)/6<1$

Similarly, we could decompose it using $1/2+1/6+...+1/n!$ whose sum converges towards $e-2<1$.

Therefore, we would never reach the end !

How can we still give an answer to Zeno's paradox with such a decomposition ?

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Sure, we never reach the end, in the sense that each partial sum is strictly less than the limit. So what? How is this any different from the usual Zeno's paradox? The answer is the same. – blue Jun 10 '14 at 11:03
@seaturtles I just thought it was weird that we never reached the end - but now i get it – Hippalectryon Jun 10 '14 at 11:04
One really ought to explain why they think something is paradoxical; while it is traditional to respond to the paradox by refuting "infinite sum = infinite time" with a convergent sum, the fact the entire sequence of events describing the travel is a transfinite (or even a more complicated order type) sequence is a much harder conceptual problem and won't be addressed by the traditional response. So if it's the latter problem you're stuck on, you generally won't get useful answers! – Hurkyl Jun 10 '14 at 11:23
Excellent point! This builds the case for my speed formula approach that I posted at the original question. – Dan Christensen Jun 10 '14 at 14:13
up vote 5 down vote accepted

Zeno's paradox tries to claim that since you need to make infinitely many steps (it does not matter which steps precisely), then it will take an infinite amount of time to get there. The mathematical solution is to sum the times and show that you get a convergent series, hence it will not take an infinite amount of time.

You show above that it is possible to split the length to travel into smaller parts in various ways. But however you will do it the series for the time it will take to traverse these smaller parts will still converge (not necessarily to the time it takes to travel the entire distance, since you may only make it to some mid-point, and then, after you covered countably many little parts, you'll need to continue traveling further). You will only recover the paradox if you could show that you can subdivide the interval in such a way that the time needed to traverse the parts totals to $\infty $. That is impossible though.

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What you're showing is just how far Zeno's paradox is from a true paradox. What the paradox tries to say is "When you traverse a unit interval, you cover half the remaining distance infinitely many times, so it must not be possible". The implied assumption is that it's not possible to do infinitely many things in the course of accomplishing a task. What you're showing is that, not only is it possible to do infinitely many things, but you may still have work left to do.

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It's easy to see that the traditional "split in half" decomposition covers the entire interval from start (inclusive) to finish (exclusive).

Why should we think the other decompositions you suggested cover the entire interval? If they do not, then we have a nonzero leftover part that happens after all of the steps you listed.

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What If i state Zeno's paradox as 'I need to cover a distance. First i cover a fourth of it, then a ninth of what remains, then a sixteenth of what remains, etc. Therefore i never reach the end' ? – Hippalectryon Jun 10 '14 at 11:00
@Hippalectryon: You don't reach the end (nor get close to it) over the part of the interval covered by that description. To get to the end, all of those infinitely many steps happen, and then there's more to do. To describe a sequence of events that includes such a decomposition and includes arriving at the endpoint will require a transfinite sequence. – Hurkyl Jun 10 '14 at 11:01

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