Suppose that $T^{⁎}$ is an ${\omega }_1$-Aronszajn tree with no stationary antichain. We introduce a forcing axiom PFA($T^{⁎}$) for proper forcings which preserve these properties of $T^{⁎}$. We prove that PFA($T^{⁎}$) implies many of the strong consequences of PFA, such as the failure of very weak club guessing, that all of the cardinal characteristics of the continuum are greater than ${\omega }_1$, and the P-ideal dichotomy. On the other hand, PFA($T^{⁎}$) implies some of the consequences of diamond principles, such as the existence of Knaster forcings which are not stationarily Knaster.

中文翻译：

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非特殊Aronszajn树的强迫公理

假设 ${Ť}^{⁎}$ 是一个 ${\omega }_{\mathrm{1个}}$-Aronszajn树，没有固定的反链。我们介绍了一个强制公理PFA（${Ť}^{⁎}$），以保留这些属性的适当强制 ${Ť}^{⁎}$。我们证明PFA（${Ť}^{⁎}$）暗示了PFA的许多强大后果，例如俱乐部非常虚弱的猜测失败，该连续统的所有基本特征都大于${\omega }_{\mathrm{1个}}$，以及P-理想二分法。另一方面，PFA（${Ť}^{⁎}$）暗示钻石原则的某些后果，例如存在不是固定Knaster的Knaster强迫。