Self-dual axioms for many-dimensional projective geometry

Proposed and compared are four equivalent sets R, S, T, D of self-dual axioms for projective geometries, using points, hyperplanes and incidence as primitive elements and relation. The set R is inductive on the number of dimensions. The sets S, T, D all include the axiom “on every n points there is a plane", the dual of this axiom, one axiom on the existence of a certain configuration, and one or several axioms on the impossibility of certain configurations. These configurations consist of ( n + 1 ) (n + 1) points and ( n + 1 ) (n + 1) planes for sets S, T, but of ( n + 2 ) (n + 2) points and ( n + 2 ) (n + 2) planes for set D. Partial results are obtained by a preliminary study of self-dual axioms for simplicial spaces (spaces which may have fewer than 3 points per line).