R Hansen says: >Lines are not ?made up? of points. When you say ?made up? you imply stacked or connected, one after the other. But as has been established here, even by some of your own posts, points cannot be stacked or connected. Thus, lines cannot be "made up" of points.
If I said a line is a "set of points" (with some further conditions) isn't tantamount to be "made up of points"? And that is more or less the standard view. All the stuff you need is in ZF along with, of course, proper definitions and development.
I think perhaps I understand your feelings on where you'd like to go with this train of thought; you'd perhaps like "line" to be a very distinct "species" of mathematical object from "point", and that in fact may be closer to the way the ancient Greeks thought about it.