TITLE: Is the theory of incidence geometry complete?
QUESTION [6 upvotes]: Consider the basic axioms of planar incidence geometry, which allow us to speak of in-betweeness, collinearity and concurrency. These axioms per se are not complete, since for example, Desargues theorem may not always hold. in fact, Desargues theorem holds if and only if the model of incidence geometry can be coordinatized by a field, i.e. KP^2 serves as a model for some field K.
My question, then, is whether the theory of planar incidence geometry together with Desargues theorem is complete? (Call this theory IG + D)
If it is not, then what time is true in RP^2 (resp CP^2) that is indepedent of the theory IG + D?

REPLY [9 votes]: Pappus theorem implies Desargues'. The theory is far from being complete, not in the logical but in the philosophical and even aesthetic sense. Why do these incidence identities look so beautiful? :) Also what about combinatorics of, say, free or projective or some other submodules of the free module over the ring? noncommutative? (There was some old activity on this subject.) What about combinatorics of geodesic surfaces in nice Riemannian manifolds, and so on?