Here is the definition of a projective plane from Stillwell's The Four Pillars of Geometry (2005, Springer):
Let $\cal P$ ("points") be a set, and let $\cal L$ ("lines") be a set of subsets of $\cal P.$ We say $({\cal P},{\cal L})$ is a projective plane to mean
- Any two distinct points lie on exactly one line;
- Any two distinct lines intersect at exactly one point; and
- There exist four points with the property that no three of these points are collinear.
Can a line have cardinality strictly less than 3? People sometimes add an axiom saying that
- Every line has cardinality at least 3. (Cf. Beck-Bleicher-Crowe's Excursions into Mathematics: The Millenium Edition (2000, CRC Press).)
I was wondering whether this fourth axiom is implied by the first three axioms. It is not extremely difficult to show that lines must have at least two points. Thus, we can reformulate the question as: Can a line have exactly two points?