Incidence Geometry
Undefined Terms: point, line, and on
Incidence Axiom 1. For each two distinct points there exists a unique line on both of them.
Incidence Axiom 2. For every line there exist at least two distinct points on it.
Incidence Axiom 3. There exist at least three distinct points.
Incidence Axiom 4. Not all points lie on the same line.
How could I add one axiom to make the set (1) inconsistent; (2) incomplete; and (3) non-independent?