# Does any real projective plane incidence theorem follow from axioms?

Is it known whether any projective geometry statement that holds true in the real projective plane (equivalently, can be deduced from Hilbert axioms) follows from the standard projective axiomatics?

By projective geometry statement I mean any statement it terms of incidence of points and lines, that is a statement of the form: Let $$I\subset \mathbb{N}\times\mathbb{N}$$ be a finite subset and $$(i_0,j_0)\in\mathbb{N}\times\mathbb{N}$$. For all sequences $$(p_i)$$ of points and sequences $$(l_j)$$ of lines, if $$p_i\in l_j$$, for all $$(i,j)\in I$$ then $$p_{i_0}\in l_{j_0}$$.

By standard projective axiomatics I mean four incidence axioms plus Desargues, Pappus and Fano.

• No. Your axioms hold in the projective plane over any field of characteristic $\neq2$. So they don't imply, for example, the existence of $\sqrt2$ in the underlying field. But that existence can be expressed as an incidence statement. Unfortunately, I don't have time right now to reconstruct that incidence statement or to search for a reference, so this is only a comment, not an answer. Jan 9, 2020 at 23:34

This is false. I don't see how to get a counterexample from Andreas Blass's comment (the only uses for the existence of $$\sqrt2$$ which are obvious to me require a more flexible notion of incidence statement), so I am posting this as an answer although it is probably more complicated than necessary.

For fixed prime $$p$$, one can take a point and line configuration corresponding to the rank 3 Dowling geometry of $$\mathbb{Z}/p$$, in which a certain equality must occur unless the underlying field has a primitive p-th root of unity. The presence of a root of unity of prime order $$p \ge 3$$ is not ruled out by these axioms, which can therefore never prove the equality, although it holds over $$\mathbb{R}$$.

Explicitly, what you do is the following: take 3 points and the 3 lines connecting them (think of this as a triangle). On each of these three lines, place $$p$$ points, labelled $$0,1,\ldots,p-1$$. Label the lines by $$a,b,c$$, and denote by $$i_a, i_b, i_c$$ the point labelled $$i$$ on the corresponding line. Define a new line on which $$i_a, j_b, k_c$$ are present whenever $$i+j+k = 0 \mod p$$.

Any realization of this point and line arrangement over a field, in which the corners of the triangle are all distinct, corresponds to a representation of $$\mathbb{Z}/p$$ in the multiplicative group of the field. Hence if $$p\ge 3$$, either some two corners coincide or the representation is trivial, so all points $$\{i_a\}_i$$ are identical, as are all points $$\{i_b\}_i$$ and all points $$\{i_c\}_i$$.

The many lines connecting various triples of points also imply that if two corners of the triangle coincide, so do all points $$i_a, j_b, k_c$$ for all $$i,j,k$$.

Hence in $$\mathbb{R}$$, one has that $$1_a$$ lies on the line through $$0_a, 0_b, 0_c$$. This incidence statement is not true over $$\mathbb{C}$$, hence it is not implied by the axioms given.

• Amazing. Thank you. The statements you give seem to be independent for different $p$'s. Is there some sensible description of real projective incidence theorems? E.g. is there finite axiomatics? Jan 10, 2020 at 10:54
• What would be a good reference to educate myself about these matters? Jan 10, 2020 at 10:58
• I learned these things by studying some matroid theory. I'm fairly sure there is no finite description in terms of incidence statements only: see Vamos, "The Missing Axiom of Matroid Theory is Lost Forever" (unfortunately not free for access as far as I can tell.) The real-representable matroids also have the property of having infinitely many excluded minors, and one can translate between your incidence statements and the language used by Vamos (if you allow conjugation and disjunction). Jan 10, 2020 at 11:09
• If you do not have access to Vámos's paper, you might want to look at a later paper by Mayhew, Newman and Whittle on the same topic. Jan 10, 2020 at 21:13