Absolute Geometry - Incompleteness

Incompleteness

Absolute geometry is an incomplete axiomatic system, in the sense that one can add extra independent axioms without making the axiom system inconsistent. One can extend absolute geometry by adding different axioms about parallel lines and get incompatible but consistent axiom systems, giving rise to Euclidean or hyperbolic geometry. Thus every theorem of absolute geometry is a theorem of hyperbolic geometry and Euclidean geometry. However the converse is not true.