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?
Best Answer
As Greg explains, the theory of projective planes obeying Desargues is basically equivalent to the theory of division rings, while the theory of projective planes obeying Desargues and Pappus as equivalent to the theory of fields. I haven't seen an axiomitization of projective planes with betweenness, but I assume that this would turn into the theory of ordered fields.
To finish the answer, one should say that none of these theories are complete. For example, the Fano plane is realizable in $KP^2$ if $K$ has characteristic $2$, but not otherwise. There is an example, which I am too lazy to draw, of an arrangement of points and lines which is true in $KP^2$ if and only if $K$ contains a square root of $5$. Thus, $\mathbb{R}P^2$ can be distinguished from $\mathbb{Q}P^2$, even though both obey Desargues and Pappus, and presumably whatever axioms of betweenness you want to impose.
You should be able to adapt the proof of Mnev's universality theorem (see also) in order to show that, if $K$ and $L$ are fields which can be distinguished by some first order property, then the projective planes $KP^2$ and $LP^2$ can be similarly distinguished.