On the axiomatics of projective and affine geometry in terms of line intersection

Abstract

By providing explicit definitions, we show that in both affine and projective geometry of dimension ≥ 3, considered as first-order theories axiomatized in terms of lines as the only variables, and the binary line-intersection predicate as primitive notion, non-intersection of two lines can be positively defined in terms of line-intersection. (2000): 51A05, 51A15, 03C40, 03B30. Mathematics Subject Classification

Coplanarity of lines in projective and polar Grassmann spaces

“…Axiomatization for higher dimensions is given in [23]. Affine spaces, except those where all lines are of size 2, can be axiomatized by means of line intersection as a sole primitive notion [13]. In view of [29] binary line intersection is sufficient to express the geometry of spine spaces.…”

Euclidean geometry of orthogonality of subspaces

“…plane Euclidean geometry as theory of tangency of circles, in [19], Euclidean and hyperbolic geometries as theories of tangency of spheres, see [11]). Among others, intersection of lines was proved to be such a notion for affine and projective geometries (of suitable dimensions, see [5], [12]). In every case a problem appears, which relations on this new universe are sufficient for the geometry in question; in particular: can we find a single sufficient binary relation?…”

Possible primitive notions for geometry of spine spaces

“…Thus the sets of the form S(a) are distinguished within C. 2For at least 3-dimensional affine spaces and for at least 4-dimensional projective spaces the result is well known as it is a result of a search for an adequate system of primitive notions for line geometry (cf [8,13,12,9,22]…”