 
Summary: Hilbert's axioms for (two dimensional) neutral geometry.
We spell these out below. It will take a while. There will be several groups of axioms: the incidence
axioms; the betweenness axioms; the continuity axiom; and the congruence axioms.
The Incidence Axioms. There are a set whose members we call points and a family of sets of points
whose members we call lines such that
(I1) if a and b are distinct points there is one and only one line
l(a, b),
the line determined by a and b, such that {a, b} l(a, b);
(I2) any line contains at least two points.
If p is a point and L is a line we say p lies on L if p L.
Theorem. If L and M are lines and L intersects M then either L = M or L M contains exactly one
point.
Proof. This follows directly from (I1).
If L and M are lines and L does not intersect M we say L and M are parallel.
Definition. Suppose S is a set of points. We say S is collinear if S is a subset of some line. We say S is
noncollinear if S is not collinear.
Note that a subset of a collinear set is collinear and that a superset of a noncollinear set is noncollinear.
(I3) There is a noncollinear set of points.
An obvious consequence of (I3) is the following.
Theorem. Suppose L is a line. Then there is a point which does not lie on L.
