Ordered Geometry - Axioms of Ordered Geometry

Axioms of Ordered Geometry

1. There exist at least two points.
2. If A and B are distinct points, there exists a C such that .
3. If, then A and C are distinct (A≠C).
4. If, then but not .
5. If C and D are distinct points on the line AB, then A is on the line CD.
6. If AB is a line, there is a point C not on the line AB.
7. (Axiom of Pasch) If ABC is a triangle and and, then there exists a point F on the line DE for which .
8. Axiom of dimensionality:
   1. For planar ordered geometry, all points are in one plane. Or
   2. If ABC is a plane, then there exists a point D not in the plane ABC.
9. All points are in the same plane, space, etc. (depending on the dimension one chooses to work within).
10. (Dedekind's Axiom) For every partition of all the points on a line into two nonempty sets such that no point of either lies between two points of the other, there is a point of one set which lies between every other point of that set and every point of the other set.

These axioms are closely related to Hilbert's axioms of order. For a comprehensive survey of axiomatizations of ordered geometry see.