关于极小非$\sigma$-散布线性序
On minimal non-$σ$-scattered linear orders
AI总结：
本文构造新的极小非$\sigma$-散布线性序，证明$\diamondsuit$蕴含极小Countryman线存在，给出基数大于$\aleph_1$的此类序的一致性例子，通过力迫和组合原理两种方法构造，并证明其在可构造宇宙及特定大基数假设下的存在性。
摘要：
本文旨在构造一类新的极小非$\sigma$-散布线性序。具体而言，我们证明Jensen原理$\diamondsuit$蕴含极小Countryman线的存在性，从而回答了Baumgartner的一个问题。我们还给出了基数大于$\aleph_1$的极小非$\sigma$-散布线性序的首个一致性例子：对任意后继基数$\kappa^+$，我们得到基数为$\kappa^+$的此类线性序，且其平方可表示为$\kappa$个链的并。我们给出两种构造方法：一是用力迫法直接构造此类例子，二是从组合原理推导其存在性。后一种方法表明，在Gödel可构造宇宙中，对任意基数$\kappa$，都存在基数为$\kappa^+$的极小非$\sigma$-散布线性序；此外（利用Rinot的工作），在不存在满足“存在Mitchell阶为$\mu^{++}$的可测基数$\mu$”的内模型的情况下，奇异强极限基数的后继处必定存在此类例子。
英文摘要：
The purpose of this article is to give new constructions of linear orders which are minimal with respect to being non-$σ$-scattered. Specifically, we will show that Jensen's principle $\diamondsuit$ implies that there is a minimal Countryman line, answering a question of Baumgartner. We also produce the first consistent examples of minimal non-$σ$-scattered linear orders of cardinality greater than $\aleph_1$, as given a successor cardinal $κ^+$, we obtain such linear orderings of cardinality $κ^+$ with the additional property that their square is the union of $κ$-many chains. We give two constructions: directly building such examples using forcing, and also deriving their existence from combinatorial principles. The latter approach shows that such minimal non-$σ$-scattered linear orders of cardinality $κ^+$ exist for every cardinal $κ$ in Gödel's constructible universe, and also (using work of Rinot) that examples must exist at successors of singular strong limit cardinals in the absence of inner models satisfying the existence of a measurable cardinal $μ$ of Mitchell order $μ^{++}$.
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