论文标题
第一个Omega Alephs：从简单到树木的树木再到更高的步行
The first omega alephs: from simplices to trees of trees to higher walks
论文作者
论文摘要
当前工作的出发点是巴里·米切尔（Barry Mitchell）1972年的定理，即$ \ aleph_n $的同谋维度为$ n+1 $。我们记录了该定理的新证明和温和的加强；不过，我们更基本的目标是澄清了较高维度的无限制组合主义者的核心。在这项工作的过程中，我们描述了列出的$ω_n$的简单特征，相干的Aronszajn树的高维概括，大型索引集的关键反向系统的基础，功能的非平凡$ n $ coherent家族以及高维步行的高维步行概括。这些构造和参数完全在$ \ Mathsf {ZFC} $ Framework中进行；他们的心是一种简单，有限的效果，可以使$ c $序列复合。
The point of departure for the present work is Barry Mitchell's 1972 theorem that the cohomological dimension of $\aleph_n$ is $n+1$. We record a new proof and mild strengthening of this theorem; our more fundamental aim, though, is some clarification of the higher-dimensional infinitary combinatorics lying at its core. In the course of this work, we describe simplicial characterizations of the ordinals $ω_n$, higher-dimensional generalizations of coherent Aronszajn trees, bases for critical inverse systems over large index sets, nontrivial $n$-coherent families of functions, and higher-dimensional generalizations of portions of Todorcevic's walks technique. These constructions and arguments are undertaken entirely within a $\mathsf{ZFC}$ framework; at their heart is a simple, finitely iterable technique of compounding $C$-sequences.
