We show that Dependent Choice is a sufficient choice principle for developing the basic theory of proper forcing, and for deriving generic absoluteness for the Chang model in the presence of large cardinals, even with respect to $\mathsf {DC}$ -preserving symmetric submodels of forcing extensions. Hence, $\mathsf {ZF}+\mathsf {DC}$ not only provides the right framework for developing classical analysis, but is also the right base theory over which to safeguard truth in analysis from the independence phenomenon in the presence of large cardinals. We also investigate some basic consequences of the Proper Forcing Axiom in $\mathsf {ZF}$ , and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in $\mathsf {ZF}+\mathsf {DC}$ and $\mathsf {ZFC}$ . Our results confirm $\mathsf {ZF} + \mathsf {DC}$ as a natural foundation for a significant portion of “classical mathematics” and provide support to the idea of this theory being also a natural foundation for a large part of set theory.

中文翻译：

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独立选择、适当性和一般绝对性

我们证明了从属选择是一个充分的选择原则，用于发展适当强迫的基本理论，并在大基数存在的情况下为 Chang 模型推导一般绝对性，即使关于$\mathsf {DC}$-保留强制扩展的对称子模型。因此，$\mathsf {ZF}+\mathsf {DC}$不仅为发展经典分析提供了正确的框架，而且还是在大基数存在的情况下保护分析真理免受独立现象影响的正确基础理论。我们还研究了适当强迫公理的一些基本结果$\mathsf {ZF}$, 并提出一个关于适当强迫公理的一般绝对性的自然问题$\mathsf {ZF}+\mathsf {DC}$和$\mathsf {ZFC}$. 我们的结果证实$\mathsf {ZF} + \mathsf {DC}$作为“经典数学”的重要部分的自然基础，并为该理论也是大部分集合论的自然基础的观点提供支持。