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 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. Collections: Mathematics