Elementary axiomatizations of projective space and of its associated Grassmann space

Abstract

It is pointed out that the axiomatizations of projective geometry by means of point-line incidence, or by means of line-intersection, and that of the Grassmann space of the lines in a projective space are mutually translatable, so that one can obtain an axiomatization of any of these theories from the other, which in particular allows us to elementarily axiomatize the Grassmann space represeting the lines of a projective space, a subject which has been extensively studied since 1981, when Tallini provided a non-elementary characterization for it.