The paradox
Take the line interval from 0 to 1. It has length 1. This line interval is composed entirely of points. Each point has zero dimensions ie no length. So the sum of the lengths of all the points in the line interval is zero.
My solution
The paradox relies on summing the lengths of all the points in the interval. Since the length of each is 0, so is their sum. My objection to this procedure is that one cannot sum over an
uncountable set. Recall that whereas the rational numbers are an infinite but countable set, the real numbers are uncountably infinite. This means that the real numbers cannot be listed sequentially as r
1, r
2, r
3, ... r
n and so on, whereas the rational numbers can.
The operation of summing anything consists of repeating the binary sequential operation called addition. The first two elements are added together, then the next element is added, and so on. This works fine for finite sums such as a restaurant bill, even if the results are often distressing. It is also OK for adding together an infinite sequence of numbers, such as 1/2, 1/4, 1/8, 1/16 and so on (the answer is exactly 1). Because the operation of summing relies on two-at-a-time addition, it is
not applicable to uncountable sets, such as the real numbers. This is because the operation of adding is sequential in character. The point is that you
can list the elements that are being added in any sum, whereas the points in a line interval cannot be listed because they are uncountable.
Unfortunately, this resolution is not intuitively satisfying, but them's the breaks.
The role of uncountability is crucial. This is demonstrated in another version of the same paradox:
Splitting a grain of sand