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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?

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There are interesting differences between finite projective planes and infinite projective planes. In the finite plane world often attempts are made to find analogues of phenomena that occur in the infinite case. – Joseph Malkevitch Dec 18 '09 at 5:04
up vote 5 down vote accepted

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.

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There is a small example of an arrangement of lines which requires a quadratic surd in the field in Grünbaum's book on convex polytopes. Google book does not have the page, afaict. – Mariano Suárez-Alvarez Dec 18 '09 at 3:45
Yeah, and it's also missing from the google books version of Ziegler's book. Page 173, for those who own a physical copy. – David Speyer Dec 18 '09 at 3:55
What page in Grunbaum's Convex Polytopes (1st edition)? Pg. 173 of Ziegler's Lectures on Polytopes, 1st edition? – Joseph Malkevitch Dec 18 '09 at 4:07

Generally Pappus theorem implies Desargues'. The theory is far from being complete not in logical but in philosophical and even aesthetics meaning. Why this incidence identities looks so beautiful? :) Also what about combinatorics of say free or ... projective or ... submodules of free module over the ring? noncommutative? (there was some old activity on this subject) What about combinatorics of geodesic surfaces in nice Rimanien manifold etc etc...

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Wikipedia says that it is a theorem of Hilbert that any projective plane that satisfies Desargues' theorem is the projective plane which is the set of lines thruogh the origin in $D^3$, where $D$ is a division ring. As Wikipedia also explains, you also need Pappus' theorem to know that $D$ is commutative. Then you are more or less done.

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Adding to Greg's and David's answers, with something more about order, seeing it is part of the question:

A projective plane that satisfies the incidence and betweenness axioms (usually in terms of a 4-ary separation relation (AB,CD) that says whether C and D are separated by A and B, if you insist on projective instead of affine planes) and Desargues' Theorem is a projective plane over an ordered skew field. Hilbert knew the ordered story too.

(I prefer the term skew field above division ring, because for some people the term division ring means it has to be finite dimensional over its centre. By the way, the first example of a skew field infinite dimensional over its centre was ordered, and was found by Hilbert in the 1903 edition of his Foundations of Geometry. So that settled completeness: there's more than one model of ordered incidence geometry.)

Two more remarks.

  1. The Artin-Schreier theorem as generalized by Pick and Szele says that a skew field can be ordered if and only if it is formally real: -1 is not a sum of products of squares. (See e.g. Chapter 6 of T.Y. Lam, A First Course in Noncommutative Rings, Springer-Verlag, New York, 1991.)

  2. My favourite configuration theorem is the Sylvester-Gallai Theorem: Any finite non-collinear set of points in the (ordinary) plane has a line passing through exactly two of the points. It holds in the projective plane over any ordered skew field (page 461 of Motzkin or just follow Coxeter's proof using the axioms of incidence and order). It is false for any skew field of finite characteristic (just consider the points coordinatized by the prime subfield), but then of course you don't have order anymore. The following question is open, and comes from L. M. Kelly: Is there an example of a skew field such that the Sylvester-Gallai Theorem holds in the projective plane over it, yet it cannot be ordered?

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