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 $({\pi }^n:4⩽n⩽\omega )$ 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 $({\pi }^n:4⩽n⩽\omega )$ 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 ($4⩽n⩽\omega )$ are strongly type-homogeneous.

我们证明了开放射影平面的理论是完整且严格稳定的，并由此推断马歇尔·霍尔的自由射影平面 $（{\pi }^{ñ}：4⩽ñ⩽\omega ）$都是基本等价的，它们的共同理论是严格稳定且可判定的，实际上是开放射影平面的理论。我们进一步描述了开放投影平面类中的基本子结构关系，并特别表明$（{\pi }^{ñ}：4⩽ñ⩽\omega ）$是基本链。然后，我们证明开放射影平面的理论没有素数模型，没有存在量公式的布尔组合的量词的消除，并且模型还不完整。最后，我们在理论模型中刻画了分叉独立性关系，并证明了${\pi }^{ñ}$的（$4⩽ñ⩽\omega ）$ 是高度同质的。