Editing Projective geometry (section)

=== Axioms using a ternary relation ===
One can pursue axiomatization by postulating a ternary relation, [ABC] to denote when three points (not all necessarily distinct) are collinear. An axiomatization may be written down in terms of this relation as well:
* C0: [ABA]
* C1: If A and B are distinct points such that [ABC] and [ABD] then [BDC]
* C2: If A and B are distinct points then there exists a third distinct point C such that [ABC]
* C3: If A and C are distinct points, and B and D are distinct points, with [BCE] and [ADE] but not [ABE], then there is a point F such that [ACF] and [BDF].
For two distinct points, A and B, the line AB is defined as consisting of all points C for which [ABC]. The axioms C0 and C1 then provide a formalization of G2; C2 for G1 and C3 for G3.

The concept of line generalizes to planes and higher-dimensional subspaces. A subspace, AB...XY may thus be recursively defined in terms of the subspace AB...X as that containing all the points of all lines YZ, as Z ranges over AB...X. Collinearity then generalizes to the relation of "independence". A set {{nowrap|{{mset|A, B, ..., Z}}}} of points is independent, [AB...Z] if {{nowrap|{{mset|A, B, ..., Z}}}} is a minimal generating subset for the subspace AB...Z.

The projective axioms may be supplemented by further axioms postulating limits on the dimension of the space. The minimum dimension is determined by the existence of an independent set of the required size. For the lowest dimensions, the relevant conditions may be stated in equivalent
form as follows. A projective space is of:
* (L1) at least dimension 0 if it has at least 1 point,
* (L2) at least dimension 1 if it has at least 2 distinct points (and therefore a line),
* (L3) at least dimension 2 if it has at least 3 non-collinear points (or two lines, or a line and a point not on the line),
* (L4) at least dimension 3 if it has at least 4 non-coplanar points.

The maximum dimension may also be determined in a similar fashion. For the lowest dimensions, they take on the following forms. A projective space is of:
* (M1) at most dimension 0 if it has no more than 1 point,
* (M2) at most dimension 1 if it has no more than 1 line,
* (M3) at most dimension 2 if it has no more than 1 plane,
and so on. It is a general theorem (a consequence of axiom (3)) that all coplanar lines intersect—the very principle that projective geometry was originally intended to embody. Therefore, property (M3) may be equivalently stated that all lines intersect one another.

It is generally assumed that projective spaces are of at least dimension 2. In some cases, if the focus is on projective planes, a variant of M3 may be postulated. The axioms of (Eves 1997: 111), for instance, include (1), (2), (L3) and (M3). Axiom (3) becomes vacuously true under (M3) and is therefore not needed in this context.