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?