Hilbert's axioms
Hilbert's axioms are a set of 20 assumptions proposed by David Hilbert in 1899 in his book Grundlagen der Geometrie (tr. The Foundations of Geometry) as the foundation for a modern treatment of Euclidean geometry. Other well-known modern axiomatizations of Euclidean geometry are those of Alfred Tarski and of George Birkhoff.
The axioms
Hilbert's axiom system is constructed with six primitive notions: three primitive terms:
point;
line;
plane;
and three primitive relations:
Betweenness, a ternary relation linking points;
Lies on (Containment), three binary relations, one linking points and straight lines, one linking points and planes, and one linking straight lines and planes;
Congruence, two binary relations, one linking line segments and one linking angles, each denoted by an infix ≅.
Note that line segments, angles, and triangles may each be defined in terms of points and straight lines, using the relations of betweenness and containment. All points, straight lines, and planes in the following axioms are distinct unless otherwise stated.
