Suppose incidence geometry q where a line is a set of points, and a set of points is referred to as a plane. this geometry has the following axioms:
axiom 1: there exist three noncollinear points.
axiom 2: given any two distinct points, then there is a unique line containing them.
axiom 3: every line contains exactly three points.
axiom 4: two distinct linear intersect.
prove the following about q:
the fano plane is a model in this geometry.
axiom 3 is independent of all the other axioms.
axiom 4 is independent of the rest.
if l1 and l2 are two distinct lines, then l1 ? l2 contains a single point.
for every line, there is a point not on that line.
q has at least 7 points.
q containing 7 points has to be the fano plane.
q can’t have more than 7 points.
following all of this, any model of q is isomorphic to the fano plane.