I investigate the relationships between three hierarchies of reflection principles for a forcing class $\Gamma $ : the hierarchy of bounded forcing axioms, of $\Sigma ^1_1$ -absoluteness, and of Aronszajn tree preservation principles. The latter principle at level $\kappa $ says that whenever T is a tree of height $\omega _1$ and width $\kappa $ that does not have a branch of order type $\omega _1$ , and whenever ${\mathord {\mathbb P}}$ is a forcing notion in $\Gamma $ , then it is not the case that ${\mathord {\mathbb P}}$ forces that T has such a branch. $\Sigma ^1_1$ -absoluteness serves as an intermediary between these principles and the bounded forcing axioms. A special case of the main result is that for forcing classes that don’t add reals, the three principles at level $2^\omega $ are equivalent. Special attention is paid to certain subclasses of subcomplete forcing, since these are natural forcing classes that don’t add reals.

中文翻译：

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ARONSZAJN 树保护和有界强迫公理

我研究了一个强制类的反射原则的三个层次结构之间的关系$\伽马$：有界强迫公理的层次结构，$\西格玛^1_1$-绝对性，以及 Aronszajn 树木保护原则。后一种原则$\卡帕$说每当吨是一棵高大的树$\欧米茄_1$和宽度$\卡帕$没有订单类型的分支$\欧米茄_1$，并且无论何时${\mathord {\mathbb P}}$是一个强制的概念$\伽马$，那么情况并非如此${\mathord {\mathbb P}}$迫使吨有这么一个分支。$\西格玛^1_1$-绝对性充当这些原则和有界强迫公理之间的中介。主要结果的一个特殊情况是，对于不添加实数的强制类，级别的三个原则$2^\欧米茄$是等价的。特别注意子完全强迫的某些子类，因为这些是不添加实数的自然强迫类。