康特曼线与连续统假设
Countryman Lines and the Continuum Hypothesis
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中文总结 AI 辅助
研究与连续统假设一致的阿隆扎扬线类的类似基结果，证明任意两条康特曼线包含同构或反同构不可数子序等陈述与CH一致，还表明弱钻石原理下不存在康特曼线的二元基，证实了谢拉在CH下的猜想。
中文摘要 AI 辅助
我们探讨了与连续统假设（CH）一致的阿隆扎扬线类的类似基结果的前景。特别地，我们证明了以下每个陈述都与CH一致：任意两条康特曼线包含同构或反同构的不可数子序；对于任何相干阿隆扎扬树\(T \subseteq {}^{< \omega_1} \omega\)，滤子\(\mathcal{U}(T)\)是超滤子。第一个结果在CH背景下证实了谢拉的一个猜想，该猜想先前被证明可由恰当力迫公理推出。另一方面，弱钻石原理\(2^\omega < 2^{\omega_1}\)意味着不存在康特曼线的二元基。
英文摘要
We explore the prospect of basis-like results for the class of Aronszajn lines which are consistent with the Continuum Hypothesis (CH). In particular, we prove that each of the following statements is consistent with CH: Any two Countryman lines contain isomorphic or anti-isomorphic uncountable suborders; for any coherent Aronszajan tree $T \subseteq {}^{< ω_1} ω$, the filter $\mathcal{U}(T)$ is an ultrafilter. The first result confirms a conjecture of Shelah in the context of CH which was previously shown to follow from the Proper Forcing Axiom. On the other hand, the weak diamond principle $2^ω< 2^{ω_1}$ implies that there does not exist a two element basis for the Countryman lines.
