To be specific, are geometric lines as defined (for example) by Hilbert's axioms isomorphic to $\mathbb{R}$? On the one hand, we clearly assume this to be the case every time we speak of the number "line." But on the other hand, $\mathbb{R}$ has special elements like "0" and "1" with important properties, whereas all points on a line are equivalent.
Also, constructing the reals is quite complicated: we need to first construct integers, then define rationals, then add in Dedekind cuts or Cauchy sequences or other fancy machinery to produce the continuum. Can lines be constructed from points in a similar way, or are we forced to assert their continuum properties through axioms? What I'm really looking for is a rigorous justification for treating the real number "line" as a line.