Gallucci's axiom revisited

In this paper we propose a well-justified synthetic approach of the projective space. We define the concepts of plane and space of incidence and also the Gallucci's axiom as an axiom to our classical projective space. To this purpose we prove from our space axioms, the theorems of Desargues, Pappus, the fundamental theorem of projectivities, and the fundamental theorem of central-axial collinearities, respectively. Our building up do not use any information on analytical projective geometry, as the concept of cross-ratio and the homogeneous coordinates of points.