According to Wikipedia(http://en.wikipedia.org/wiki/Projective_geometry#Axioms_of_projective_geometry), the axioms of projective geometry due to A. N. Whitehead are:
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).
I think that every point of a projective space other than a point must belong to a line, but I wonder if we can prove it using these axioms.
Well, if you really want to work with Whitehead's axioms, here are the axioms as he put them in The axioms of projective geometry page 7:
As you can see, axiom $II$ asserts there is a point, and axiom $III$ asserts there is a point other than the first point you picked. Then axioms $IV$ and $VII$ give you that the point you picked lies on a line.
As written, none of the set of three axioms you gave asserts the existence of a point or line, so the emptyset or a single point set vacuously satisfies all three. But if you were given at least two points, then you can pick any two points and say "there's a line through them," using G2, so that every point lies on a line.
If you are learning directly from the Wikipedia article, then it's good to keep in mind that Wiki articles aren't always perfect. The description of Whitehead's axioms at that link is likely shortened in the interest of brevity. It says the axioms are "based on" Whitehead's axioms.