We investigate forcing properties of perfect tree forcings defined by Prikry to answer a question of Solovay in the late 1960's regarding first failures of distributivity. Given a strictly increasing sequence of regular cardinals $〈{\kappa }_n:n<\omega 〉$, Prikry defined the forcing $\mathbb{P}$ of all perfect subtrees of ${\prod }_{n<\omega }{\kappa }_n$, and proved that for $\kappa ={\sup }_{n<\omega }{\kappa }_n$, assuming the necessary cardinal arithmetic, the Boolean completion $\mathbb{B}$ of $\mathbb{P}$ is $(\omega ,\mu )$-distributive for all $\mu <\kappa$ but $(\omega ,\kappa ,\delta )$-distributivity fails for all $\delta <\kappa$, implying failure of the $(\omega ,\kappa )$-d.l. These hitherto unpublished results are included, setting the stage for the following recent results. $\mathbb{P}$ satisfies a Sacks-type property, implying that $\mathbb{B}$ is $(\omega ,\infty ,<\kappa )$-distributive. The $(\mathfrak{h},2)$-d.l. and the $(\mathfrak{d},\infty ,<\kappa )$-d.l. fail in $\mathbb{B}$. $\mathcal{P}(\omega )/\mathit{fin}$ completely embeds into $\mathbb{B}$. Also, $\mathbb{B}$ collapses ${\kappa }^{\omega }$ to $\mathfrak{h}$. We further prove that if κ is a limit of countably many measurable cardinals, then $\mathbb{B}$ adds a minimal degree of constructibility for new ω-sequences. Some of these results generalize to cardinals κ with uncountable cofinality.

我们调查了由Prikry定义的完美树强迫的强迫属性，以回答Solovay在1960年代后期有关首次分配失败的问题。鉴于常规红衣主教严格增加了顺序$〈{\kappa }_{ñ}：ñ<\omega 〉$，Prikry定义了强制 $\mathbb{P}$ 的所有完美子树 ${\prod }_{ñ<\omega }{\kappa }_{ñ}$，并证明 $\kappa ={\mathrm{SUP}}_{ñ<\omega }{\kappa }_{ñ}$，假设必要的基本算术，则布尔完成 $\mathbb{乙}$ 的 $\mathbb{P}$ 是 $（\omega ，\mu ）$-为所有人分配 $\mu <\kappa$ 但 $（\omega ，\kappa ，\delta ）$-所有人的分配失败 $\delta <\kappa$，表示失败 $（\omega ，\kappa ）$-dl包括了这些迄今尚未发布的结果，为以下最近的结果奠定了基础。 $\mathbb{P}$ 满足Sacks类型的属性，这意味着 $\mathbb{乙}$ 是 $（\omega ，\infty ，<\kappa ）$-分布。的$（\mathfrak{H}，2）$-dl和 $（\mathfrak{d}，\infty ，<\kappa ）$-dl失败 $\mathbb{乙}$。 $\mathcal{P}（\omega ）/\mathit{鳍}$ 完全嵌入 $\mathbb{乙}$。也，$\mathbb{乙}$ 崩溃 ${\kappa }^{\omega }$ 至 $\mathfrak{H}$。我们进一步证明，如果κ是许多可测量基数的极限，则$\mathbb{乙}$为新的ω序列增加了最小程度的可构造性。其中一些结果普遍适用于具有不可数最终确定性的红衣主教κ。