How can a set of points have infinitely many members but zero length?

Because counting points and measuring length are two different ways of saying how big a set is, and they don't have to agree. The classic example is the Cantor set. Take a segment from 0 to 1 and erase its open middle third, leaving two pieces. Erase the middle third of each, leaving four. Keep going forever. What survives has zero length, yet it holds as many points as the segment you started with.

At every step you remove length, and if you add up everything you erased, it comes out to the full length of the original segment. So what's left has zero length in the usual sense: mathematicians say it has 'measure zero'. And yet, astonishingly, the Cantor set is uncountable, meaning it contains just as many points as the entire line segment you started with. A set can take up no space at all and still contain infinitely many, unlistably many, points.

The set is also self-similar: it looks exactly like two shrunken copies of itself, one squeezed into the left third of the line and one into the right third. That kind of structure, which repeats at every scale no matter how far you zoom in, makes the Cantor set the prototype example of what mathematicians now call a fractal.

It was discovered in 1874 by Henry John Stephen Smith, though it carries the name of Georg Cantor, who mentioned it in passing in 1883; by studying sets like it, Cantor and others helped lay the foundations of modern point-set topology. The set's own name is a quiet reminder that credit in mathematics doesn't always go to the first person who found something.