Fun with a dichotomy paradox

This is a variation of Zeno of Elea's dichotomy paradox.

Let's suppose that there is a finite distance, and one must travel from one extrema (point A) to the other extrema (point B). However, because the distance is infinitely divisible, one must traverse an infinite number of points in order to reach the other extrema, thus making the finite distance infinite.

Or for clarity:

(1) The distance between point A and point B is a finite distance.
(2) Measured distance is infinitely divisible.
(3) An infinitely divided distance consists of an infinite number of points.
(4) A distance consisting of an infinite number of points is infinite.

One could say that therefore space (unlike mathematical measurement) isn't infinitely divisible, but I feel as though that would be a lazy way to go about it..

The argument begins with the statement that the distance between A and B is finite, and reaches a conclusion that that same distance is infinite.

Perhaps because the infinite number of points lies within the constraints of two extremes (A and B), the distance remains finite, though there are an infinite number of points between them... although that in itself is paradoxical.

It looks like the trouble stems from some misconception or misapplication of the concept of infinite divisibility, but at the same time all of the premises sound perfectly logical. I guess that's why it's a paradox.

Or.. perhaps (4) is false: a distance consisting of an infinite number of points isn't necessarily of an infinite length? But I'm not sure how one would go about qualifying that statement.

The popular response to the argument is to pace back and forth and not utter a single word, but that merely demonstrates that it's empirically disproven, it doesn't point out which of the premises is invalid.

1. Why Mathematical Solutions of Zeno's Paradoxes Miss the Point


10/21/06 update:

I've been putting this off for a while, but here's my current opinion on this paradox: It is presupposed that we have a distance with two termini, points A and B. This distance can said to be subdivided indefinitely, creating an infinite number of points within the domain of a finite distance... this doesn't demonstrate that space cannot be infinitely subdivided, but rather it presents a paradox, the congruence of two incompatible things... nor does it disprove that there is motion— that is an a posteriori phenomena which is presupposed. Rather— granting that we have motion, the distance between two points still cannot be traversed if the distance is ultimately infinite.

Now, let's look at it this way: we will subdivide this distance at regular intervals, so:

A---|---|---|---|---|---B

Let's now say that, with motion presupposed, it would take 1 second from one marker to the next (A and B are markers also).

And then the line is again divided:

A-}-|-}-|-}-|-}-|-}-|-}-B

Here the distance between |'s is 1 second and }'s is 0.5. It's important to bear in mind the rate in which the infinite subdivision occurs. One could assert that between every 1 second, .5 second, .25 second, etc. division there is an infinite number of points and so respectively the distance could be mathematically written as 6*infinity, 12*infinity, 24*infinity, etc. etc. The important thing to remember is the rate at which the infinite subdivision occurs (i.e. the intervals of subdivision). Eventually there may be a subdivision of 0.5^{-1,000,000,000,000}, but the time required to carry between markers (with motion again presupposed) would be infinitesimally small and, though composed of an infinite sum of time-intervals, still comprise a finite distance (e.g. between markers, the motion-time-thing would remain 1).