How to prove that any line contain at least three points?

Question

Hi i was reading a book called Symmetry and Pattern in Projective Geometry by Eric Lord, in his book the author give these axioms:

1. Any two distinct points are contained in a unique line.
2. In any plane, any two distinct lines contain a unique common point.
3. Three points that do not lie on one line are contained in a unique plane.
4. Three planes that do not contain a common line contain a unique common point.

My question is if with these axioms can i prove the statement that any line contains at least three points?

Answer 1

As far as I can see, a line with two points satisfies this system of axioms.

None of these axioms postulate the existence of noncollinear points, but that is normally a feature of axioms for the projective plane and projective $3$-space.

Perhaps the author has given these axioms in addition to some others that occurred earlier?

Answer 2

If we're allowed to use this definition for a line in $\mathbb{R}^{3}$:

$L = \vec{a} + \lambda \vec{u}: \lambda \in \mathbb{R}$, $\vec{a}, \vec{u} \in \mathbb{R}^{3}$

Where $\vec{a}$ and $\vec{u}$ are two distinct points contained by $L$

Then by changing the value of $\lambda$ we can show that $L$ contains at least $3$ points. Although, this definition doesn't come straight from those axioms, so it seems like you can't prove the statement using only those axioms.