Axiom of Parallels

The axiom in this section caused the most controversy and confusion of all. The axioms of parallels (which is also an incidence axiom) is

Axiom of Parallels

Given a line and a point outside it there is exactly one line through the given point which lies in the plane of the given line and point so that the two lines do not meet.

Note that, while asserting that there is a line through the given point that doesn't meet the given line, it also says there is only one such line. In other words, it also asserts that all the ``other'' lines co-planar with the given line meet that line. This motivates the introduction of the following (stronger and stranger) version of the Axiom of Parallels:

Projective Axiom of Parallels

Any pair of lines that lie in the same plane meet.

The idea behind this axiom is that even (apparently) parallel lines appear to meet at the horizon. We can demonstrate that this axiom is consistent with the axioms of Incidence by means of Linear Algebra as in the examples below.