G1: Every line contains at least 3 points
G2: Every two points, A and B, lie on a unique line, AB.
G3: If lines AB and CD intersect, then so do lines AC and BD
(where it is assumed that A and D are distinct from B and C).
Now let $V$ be a vector space over a field $K$.
We denote by $P(V)$ the set of one-dimensional subspaces of $V$.
If $V$ is finite dimensional, this is the usual definition of a projective space over $K$.
We say a two-dimensional subspace of $V$ a line of $P(V)$.
Then it is natural to expect that points and lines of $P(V)$ satisfy the above axioms.
I think it is easy to prove G1 and G2.
But how can we prove G3?
Claim: $AB$ intersects $CD$ if and only if the underlying vectors of $A,B,C,D$ are linearly dependent.
(in fact, if $AB \neq CD$, they span a space of dimension $3$)