依赖选择、恰当性与泛型绝对性
Dependent Choice, Properness, and Generic Absoluteness
AI总结：
本文证明依赖选择原理足以发展恰当力迫理论，并在大基数下保证Chang模型的泛型绝对性，从而确立ZF+DC作为经典数学和集合论重要部分的自然基础。
摘要：
我们证明，依赖选择是发展恰当力迫基本理论的充分选择原理，并且在大基数存在的情况下，即使对于力迫扩张的保持DC的对称子模型，也能推导出Chang模型的泛型绝对性。因此，ZF+DC不仅为发展经典分析提供了正确的框架，而且也是在大基数存在的情况下，保护分析中的真理免受独立性现象影响的正确基础理论。我们还研究了ZF中恰当力迫公理的一些基本推论，并提出了一个关于ZF+DC和ZFC中恰当力迫公理的泛型绝对性的自然问题。我们的结果证实ZF+DC是“经典数学”重要部分的自然基础，并支持该理论也是集合论很大一部分的自然基础这一观点。
英文摘要：
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 DC-preserving symmetric submodels of forcing extensions. Hence, ZF+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 ZF, and formulate a natural question about the generic absoluteness of the Proper Forcing Axiom in ZF+DC and ZFC. Our results confirm ZF+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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
