What is the answer to the paradox of the infinitesimal? I just read this article on npr, which mentioned the following question:


  You can keep on dividing forever, so every line has an infinite amount
  of parts. But how long are those parts? If they're anything greater
  than zero, then the line would seem to be infinitely long. And if
  they're zero, well, then no matter how many parts there are, the
  length of the line would still be zero.


It further mentions that


  Today, mathematicians have found ways to answer that question so that
  modern calculus is rigorous and reliable.


Can anyone elaborate on the modern answers to this question?
The paradox has to do with the additivity with measure; in particular, naively switching from finite additivity to infinite additivity.

We're familiar with the fact that, if we split something up into two parts, the measure of the whole is the sum of the measure of the two parts.

If we repeat this with one of the individual parts, we've now split the original whole into three parts. The measure of the whole is the sum of the individual measures of the three parts.

And so forth; binary additivity of measure does extend to arbitrary, but finite additivity of measure.

Generally speaking, things have to change when you switch from finite to infinite. We can no longer justify additivity of measure when you have infinitely many parts, because you can never get to infinitely many parts by repeatedly splitting the whole into finitely many parts finitely many times.

A priori, there might not even be a reasonable notion of additivity of measure when you have infinitely many parts! However, experience has shown there is a useful extension of additivity to countably many parts, at least when studying a continuum.

i.e. if you split a whole into countably many parts (and in a measurable way), you can expect the measure of the whole to be the sum of the measures of the individual parts.

Note that "sum" must be meant in the sense of an infinite sum from calculus; e.g. as a limit of partial sums. Trying to literally interpret an infinite sum as repeated addition runs into all of the same problems we're trying to work around.

When you split the whole into more than countably many parts -- e.g. you split the number line into its individual points -- you now have too many parts for countable additivity of measure.
It states 


  ...then the line would seem to be infinitely long.


It is infinitely long for a very very small person who walk exactly one real number with each step. For example if she/he is on the point 0.5, then she/he must find the next real point right after (or before) 0.5. Can she/he?

But for us, who use centimeter as a measuring unit, that line would not be infinitely long. So it basically depends on the unit of length that we consider.