First-Order Model Theory of Free Projective Planes

Abstract

We prove that the theory of open projective planes is complete and strictly stable, and infer from this that Marshall Hall's free projective planes $(\pi^n : 4 \leq n \leq \omega)$ are all elementary equivalent and that their common theory is strictly stable and decidable, being in fact the theory of open projective planes. We further characterize the elementary substructure relation in the class of open projective planes, and show in particular that $(\pi^n : 4 \leq n \leq \omega)$ is an elementary chain. We then prove that the theory of open projective planes does not have a prime model, that it has elimination of quantifiers down to Boolean combinations of existential formulas, and that it is not model complete. Finally, we characterize the forking independence relation in models of the theory and prove that the $\pi^n$'s ($4 \leq n \leq \omega)$ are strongly type-homogeneous.