In the first part of the article, we show that if $\omega \le \kappa < \lambda$ are cardinals, ${\kappa ^{ < \kappa }} = \kappa$, and λ is weakly compact, then in $V\left[M {\left( {\kappa ,\lambda } \right)} \right]$ the tree property at $$\lambda = \left( {\kappa ^{ + + } } \right)^{V\left[ {\left( {\kappa ,\lambda } \right)} \right]} $$ is indestructible under all ${\kappa ^ + }$-cc forcing notions which live in $V\left[ {{\rm{Add}}\left( {\kappa ,\lambda } \right)} \right]$, where ${\rm{Add}}\left( {\kappa ,\lambda } \right)$ is the Cohen forcing for adding λ-many subsets of κ and $\left( {\kappa ,\lambda } \right)$ is the standard Mitchell forcing for obtaining the tree property at $\lambda = \left( {\kappa ^{ + + } } \right)^{V\left[ {\left( {\kappa ,\lambda } \right)} \right]} $. This result has direct applications to Prikry-type forcing notions and generalized cardinal invariants. In the second part, we assume that λ is supercompact and generalize the construction and obtain a model ${V^{\rm{*}}}$, a generic extension of V, in which the tree property at ${\left( {{\kappa ^{ + + }}} \right)^{{V^{\rm{*}}}}}$ is indestructible under all ${\kappa ^ + }$-cc forcing notions living in $V\left[ {{\rm{Add}}\left( {\kappa ,\lambda } \right)} \right]$, and in addition under all forcing notions living in ${V^{\rm{*}}}$ which are ${\kappa ^ + }$-closed and “liftable” in a prescribed sense (such as ${\kappa ^{ + + }}$-directed closed forcings or well-met forcings which are ${\kappa ^{ + + }}$-closed with the greatest lower bounds).

中文翻译：

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树属性的不可破坏性

在文章的第一部分，我们展示了如果$\omega \le \kappa < \lambda$是红衣主教，${\kappa ^{ < \kappa }} = \kappa$， 和λ是弱紧致的，那么在$V\left[M {\left( {\kappa ,\lambda } \right)} \right]$树属性在$$\lambda = \left( {\kappa ^{ + + } } \right)^{V\left[ {\left( {\kappa ,\lambda } \right)} \right]} $$是坚不可摧的${\kappa ^ + }$-cc 强制存在的概念$V\left[ {{\rm{加}}\left( {\kappa ,\lambda } \right)} \right]$， 在哪里${\rm{加}}\left( {\kappa ,\lambda } \right)$是添加的 Cohen 强迫λ- 许多子集κ和$\left( {\kappa ,\lambda } \right)$是获取树属性的标准米切尔强迫$\lambda = \left( {\kappa ^{ + + } } \right)^{V\left[ {\left( {\kappa ,\lambda } \right)} \right]} $. 该结果直接应用于 Prikry 型强迫概念和广义基数不变量。在第二部分，我们假设λ是超紧的，并泛化构造并获得模型${V^{\rm{*}}}$, 的一般扩展五，其中树属性在${\left( {{\kappa ^{ + + }}} \right)^{{V^{\rm{*}}}}}$是坚不可摧的${\kappa ^ + }$-cc 强迫观念生活在$V\left[ {{\rm{加}}\left( {\kappa ,\lambda } \right)} \right]$，此外，在所有强迫观念下，生活在${V^{\rm{*}}}$哪个是${\kappa ^ + }$-在规定的意义上是封闭的和“可提升的”（例如${\kappa ^{ + + }}$- 有向的闭合强迫或满足的强迫，它们是${\kappa ^{ + + }}$- 以最大下限闭合）。