AI 中文总结
该研究从大基数出发，通过力迫构造证明了ITP可在奇异基数的可数初始段后继处同时成立，并推广得到奇异基数多共尾度长段后继处的强树性质的一致性。
AI 中文摘要
强树性质与超树性质（又称ITP）是树性质的推广，分别刻画了不可达基数下的强紧致性与超紧致性：不可达基数κ是强紧致的当且仅当κ处成立强树性质，是超紧致的当且仅当κ处成立ITP。推广Golshani与Hayut的结果，我们证明从大基数出发，ITP在奇异基数的任意可数初始段后继处同时成立是一致的；更正式地，对任意可数序数θ，我们构造了一个力迫扩张，使得ITP在奇异基数的前θ个后继处成立。我们还将该结果进一步推广，以同时获得具有多种共尾度的奇异基数的长段后继处的强树性质。
英文摘要
The strong tree property and the super tree property (also called ITP) 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 $κ$. Generalizing a result of Golshani and Hayut, we show that from large cardinals it is consistent for ITP to hold simultaneously at any countable initial segment of successors of singular cardinals. More formally, given any countable ordinal $θ$, we construct a forcing extension in which ITP holds at the first $θ$ successors of singulars. We then extend this result further to obtain the strong tree property on long segments of successors of singular cardinals of multiple cofinalities simultaneously.
Comments16 pages
