TITLE: Why do all incidence theorems follow from Pappus' theorem?
QUESTION [19 upvotes]: In Hilbert and Cohn-Vossen's ``Geometry and the Imagination,"
they state in the last paragraph of Chapter 20 that "Any 
theorems concerned solely with incidence relations in the 
[Euclidean projective] plane can be derived from [Pappus' 
Theorem]."
Does anyone know a reference for more details about this claim?
Does anyone have an idea of how to prove it?
I have seen two books by John Stillwell:  ``Four Pillars of 
Geometry" and "Yearning for the Impossible."  These books 
discuss the relation between Pappus' theorem in a projective
plane over some algebraic structure and commutativity of the 
multiplication in that algebraic structure.  In particular 
Stillwell describes how all the laws of algebra follow from
Pappus' theorem.
Presumably one then argues that any incidence theorem can be
proved using algebra.  I would be happy, for instance, with a
precise statement like this, including in particular a 
definition of ``incidence theorem."

REPLY [10 votes]: All incidence theorems, which hold over any field, follow from Pappus, because Pappus allows to sum up and multiple points on a line, i.e. implies that your plane is over field (Desargues implies that the plane is over skew field). But some specific properties of the field may be encoded as incidence theorems (I think, any algebraic property like $x^2+1=0$ has no solutions'' may be encoded, just use the geometric construction for addition and multiplication), and hold not for all fields, see spectacular examples in David's answer.