论文标题
奇异枢机主教继任者的坚固而超级树的财产
The Strong and Super Tree Property at Successors of Singular Cardinals
论文作者
论文摘要
强的树特性和ITP（也称为超级树特性）是树的概括，这些属性表征了坚固的紧凑性和超紧张性，直到无法获得性。也就是说，当且仅当强大的树木财产以$κ$为单位时，并且仅当ITP持有$κ$时，就无法访问的红衣主教$κ$非常紧凑。我们提出了几个结果，这些结果是在许多连续的红衣主教同时获得强大的树木特性和ITP的问题。这些结果集中在奇异红衣主教的继任者上。我们描述了一类强迫阶层，该类别将在任何辅助性的奇异基础主教的继任者中获得强大的树种和ITP。概括了Neeman关于Tree属性的结果，我们表明，ITP以$ \ aleph_n $的持有$ 2 \ leq n <ω$，同时使用$ \ aleph_ {ω+1} $同时持有$ \ aleph_n $。我们还表明，ITP以$ \ aleph_n $的所有$ 3 <n <ω$和$ \ aleph_ {ω+1} $同时持有以$ \ aleph_n $的形式保持一致。最后，将我们的注意力转向了无法数量的辅助性的奇异红衣主教，我们表明，强大而超级树的特性同时在多个共同性的奇异人的继任者中保持一致。
The strong tree property and ITP (also called the super tree property) are generalizations of the tree property that characterize strong compactness and supercompactness up to inaccessibility. That is, an inaccessible cardinal $κ$ is strongly compact if and only if the strong tree property holds at $κ$, and supercompact if and only if ITP holds at $κ$. We present several results motivated by the problem of obtaining the strong tree property and ITP at many successive cardinals simultaneously; these results focus on the successors of singular cardinals. We describe a general class of forcings that will obtain the strong tree property and ITP at the successor of a singular cardinal of any cofinality. Generalizing a result of Neeman about the tree property, we show that it is consistent for ITP to hold at $\aleph_n$ for all $2 \leq n < ω$ simultaneously with the strong tree property at $\aleph_{ω+1}$; we also show that it is consistent for ITP to hold at $\aleph_n$ for all $3 < n < ω$ and at $\aleph_{ω+1}$ simultaneously. Finally, turning our attention to singular cardinals of uncountable cofinality, we show that it is consistent for the strong and super tree properties to hold at successors of singulars of multiple cofinalities simultaneously.
