论文标题
Aronszajn免费kurepa树
Aronszajn Free Kurepa Trees
论文作者
论文摘要
我们考虑了$ω_1$的功率集的传递关系，并显示如果相对于此关系有最大元素，则有一个没有Aronszajn子树的kurepa树。我们还表明，如果有$ω_1$的最大子集，那么有kurepa树不是俱乐部同构。这些最大的$ω_1$的最大子集存在于许多已知模型中，这些模型是从构造宇宙中获得的，没有大型基本假设。例如，每当每当ω_1$和$ x \ subsetω_1$中$α_0\时$ \ textsc {l} [x] $ of $ x $在$ \ textsc {v} $中是最大的。
We consider a transitive relation on the power set of $ω_1$ and show if there is a maximal element with respect to this relation then there is a Kurepa tree with no Aronszajn subtree. We also show that if there is a maximal subset of $ω_1$, then there are Kurepa trees which are not club isomorphic. These maximal subsets of $ω_1$ exist in many known models that are obtained from the constructible universe without large cardinal assumptions. For instance, whenever $α_0 \in ω_1$ and $X \subset ω_1$ are such that $ω_1^{\textsc{L}[X \cap α_0]} = ω_1, ω_2^{\textsc{L}[X]} = ω_2$ and $\textsc{V}$ is a semiproper forcing extension of $\textsc{L}[X]$ then $X$ is maximal in $\textsc{V}$.
