The axioms for the Pappus geometry are:
There exists at least one line.
Every line has exactly three points.
Not all lines are on the same point. [N.B. Change from the text]
If a point is not on a given line, then there exists exactly one line on the point that is parallel to the given line.
If P is a point not on a line, there exists exactly one point P' on the line such that no line joins P and P'.
With the exception in Axiom 5, if P and Q are distinct points, then exactly one line contains both of them.
The following is a proof that the geometry of Pappus contains exactly nine points, but the sentences are out of order. Indicate the correct order of these sentences.
__6___ Plus the 3 on l, gives 9 points in total.
__2___ Every point not on l is joined to two of these three points by lines (Axioms 5 and 6).
__3___ Each of these lines has two points not on l.
__1___ There is a line, l, (Axiom 1), with three points on it (Axiom 2).
__5___ This gives 6(2) = 12 possible points, but we have counted each point twice, so there are only 12/2 = 6 points not on l.
__4___ Since there are three lines through a point (Thm 1.10), there are exactly 6 lines, other than l, that go through points of l.
Statements labeled 2, 3 and 4 above can be written in any order as long as 3 is not first.