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

TL;DR

This paper demonstrates that in affine and projective geometries of dimension three or higher, non-intersection of lines can be explicitly defined using only line intersection as a primitive concept within first-order theories.

Contribution

It provides explicit definitions showing non-intersection can be positively defined solely through line intersection in these geometries.

Findings

Non-intersection is positively definable in affine and projective geometries of dimension ≥ 3.

Line intersection alone suffices as a primitive in first-order axiomatizations.

The results clarify the logical foundations of geometric theories.

Abstract

By providing explicit definitions, we show that in both affine and projective geometry of dimension $\geq 3$, 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.