Covering with a Convergent Canopy

In the item Some Musings On Mathematics we saw how the real numbers seem to emerge from a natural succession of questions, and then in Trapping a Transcendental we saw how even the algebraic numbers are insufficient to complete the real line. We saw how to construct an increasing sequence of rationals that was bounded above. We would naturally want such a sequence to have a limit, and the algebraics don't suffice.

In this installment we'll use the techniques developed to derive an apparent paradox.

We start by enumerating the rationals. Having done that we have our fractions listed in order, every fraction appearing in our list. The specific technique given elsewhere gives this ordering:

but the exact ordering doesn't matter.

Now let's take some line segments. We take a line segment of length 1/2 and centre it on our first rational. In our specific case that means we take the segment [-1/4,1/4]. Then we take a line segment of length 1/4 and centre it on our second rational. In our specific case that means we take the line segment [-9/8,-7/8].

And so we continue. Each time we take a segment half the size and centre it on the next rational.

Observe that the endpoints of each segment are themselves rationals, and so all the segments overlap with each other. It would appear that the entire numberline must eventually get covered in this manner.

And yet the total length of line segments is 1/2+1/4+1/8+1/16+... which comes to 1. How can we cover the entire real line with line segments of total length 1?

Clearly we can't. The challenge is to understand what doesn't get covered, and why.