But if points have zero extension then even an infinity of them cannot sum to anything greater than zero.
Can you make this rigorous? Because using the standard definitions, this statement is not true. It's true that a countable number of points must have total length zero (and you can even give a rigorous proof of this) but not necessarily true for a non-countable number of points. The study of "lengths of sets of points" is called measure theory.
I think it is unnecessary, however, to bring in the whole concept of length when defining lines. For example, we could simply define a line as a set of points obeying some special properties.
Comments
Can you make this rigorous? Because using the standard definitions, this statement is not true. It's true that a countable number of points must have total length zero (and you can even give a rigorous proof of this) but not necessarily true for a non-countable number of points. The study of "lengths of sets of points" is called measure theory.
I think it is unnecessary, however, to bring in the whole concept of length when defining lines. For example, we could simply define a line as a set of points obeying some special properties.