This question arises probably from my astonishingly poor knowledge of classical euclidean geometry's axioms.
Let $ l $ be a line. I have to admit I never understood how we can say that $ l $ is in one-to-one correspondence with the real line (as some texbooks write). As last premise, I should also say that I never taken any formal logic/MT course.
Actually, we can build a bijection between a subset $ \left\{P + tv\right\}_{t\in\mathbb R} $ of $ \mathbb R^2 $ and $ \mathbb R $, in a straightforward way; in other words, we can say that some objects of a model of euclidean geometry can be putted in one-to-one correspondence with $ \mathbb R $.
My question now is: In classical axiomatic geometry, is it an axiom that given a line $ l $, and two points $ O $ and $ A $ laying on $ l $, there is a unique "order relation", "ordering", such that $ O $ precedes $ A $? Or is the existence of such a relation (to be intended as a predicate in two variables, I think) precedes actually given as an axiom, and its uniqueness can be deduced from other stuff?
And how about the $ \mathbb R^n $ version of this statement? Given a set of the form of $ \left\{P + tv\right\}_{t\in\mathbb R} $ (what in my mind and in Apostol's calculus book is called a line), and two points $ O,A\in l $, what can we say about putting an order relation on $ l $? From my point of view, the mapping $ t\mapsto O + t(A-O) $ induces an ordering $ {\leqq_{(O,A)}} $ on $ l $, such that $ O\leqq_{(O,A)}A $; but in what sense should I think about that "uniqueness"?