Projective-type axioms for the hyperbolic plane
Karl Menger showed in 1938 that hyperbolic geometry can be developed in terms of the primitive notions of ``point'', ``line'' and ``incidence'' along with postulates that refer only to collinearity, concurrency and the existence of points and lines. However, one of these postulates assumes a special case of the Fundamental Law of Projectivities and so brings in set-theoretical and inductive considerations not present in any of the other axioms. The author shows that the laws of Pappus and Desargues, restricted to cases when the relevant lines intersect, imply the Fundamental Law of Projectivities in the hyperbolic plane. The author thus shows that hyperbolic geometry can be derived solely from configuration postulates.
scientific article; zbMATH DE number 147301