What if you re-arrange axioms, so first look at I.2 then I.1 and then I.3.Originally Posted by topsquark
I.2 is there to insure existence of line as an element, so if we begin with that axiom we insure that there is a line if there are two points whether they are distinct or not, even if its the same point then through one point there is also at least one line. If this axiom wouldn't be then line wouldn't even exist at all!
I.1 states that every line has two distinct points so that axiom determines line in space.
I.3 insures that every line that has two distinct points is unique line.