Hilbert's Axioms - The Axioms

The Axioms

Hilbert's axiom system is constructed with nine primitive notions: three primitive terms:

• point;
• straight line;
• plane;

and six primitive relations:

• Betweenness, a ternary relation linking points;
• 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.