That's the thing though. As I understand it, or as it's said to be: points have zero extension. So, no amount of points, not even an infinity of them could ever have extension. But, it should make sense for a line to be composed of entities with infinitesimal extension as you say. I have seen the term linelet used before for these entities.
Charles Sanders Peirce said in 1903, ”Now if we are to accept the common idea of continuity […] we must either say that a continuous line contains no points or […] that the principle of excluded middle does not hold of these points. The principle of excluded middle applies only to an individual […] but places being mere possibilities without actual existence are not individuals.”
> That's the thing though. As I understand it, or as it's said to be: points have zero extension. So, no amount of points, not even an infinity of them could ever have extension.
This isn't true for an uncountably infinite set of points, assuming by 'extension' you mean what is usually called 'meausre' in modern mathematics. Modern theory is perfectly fine with saying that a line of nonzero length contains an infinite number of points of zero length, and trying to draw on Euclidean notions definitions of 'point' and 'line' to find conclusions about real analysis is going to be unhelpful.
I'm not sure what that Peirce quote is trying to say.
Charles Sanders Peirce said in 1903, ”Now if we are to accept the common idea of continuity […] we must either say that a continuous line contains no points or […] that the principle of excluded middle does not hold of these points. The principle of excluded middle applies only to an individual […] but places being mere possibilities without actual existence are not individuals.”