AI 中文总结
本文针对无选择公理下的可数链条件，证明若其有限支持迭代仍成立则需可数可数集族选择公理，修正相关定义问题，同时研究Knaster原理与该条件的关系。
AI 中文摘要
文献[10]提出，在无选择公理（$\boldsymbol{\text{ZF}}$）的情况下，可数链条件应被定义为“每个预稠密集都包含一个可数预稠密子集”。Philipp Schlicht在2024年“选择公理120周年”会议的教程中指出，该定义在$\text{ZF}$中不存在迭代定理，本文对此结论给出证明。具体而言，我们证明若所有ccc力迫的有限支持迭代仍为ccc，则可数可数集族的选择公理必须成立；同时，我们证明在依赖选择公理（Principle of Dependent Choice）的假设下，ccc力迫的有限支持迭代仍为ccc，这需要修正文献[1]中恰当性（properness）定义（以及Mekler的ccc定义）的一些问题。最后，我们研究Knaster原理及其与该版本ccc的关系。
英文摘要
In [10] it was suggested that the countable chain condition should be defined in the absence of choice as "every predense set contains a countable predense subset". During his tutorial at the "120 Years of Choice" conference in 2024, Philipp Schlicht remarked that this definition does not have an iteration theorem in $\mathsf{ZF}$. We provide a proof of this claim. Specifically, we show that if every finite support iteration of ccc forcings is again ccc, then the Axiom of Choice for countable families of countable sets must hold. However, we prove that under the assumption of the Principle of Dependent Choice, the finite support iteration of ccc forcings is again ccc. This requires correcting some issues with the definition of properness (and therefore Mekler's definition of ccc) from [1]. We then study Knaster principles and their relations to this version of ccc.
Comments12 pages
