A long standing project in set theory is to analyze how much compactness can be obtained in the universe. Compactness is the phenomenon where if a certain property holds for all small substructures of an object, then it holds for the entire object. Compactness properties of particular interest are combinatorial principles that follow from large cardinals, but can be forced to hold at successors. Key examples include (in order of increasing strength) failure of squares, the tree property, and the ineffable tree property (ITP). These principles “capture” the combinatorial essence of certain large cardinals. At an inaccessible cardinal, the tree property is equivalent to weak compactness; ITP is equivalent to supercompactness. Forcing these principles at successors tells us to what extent small cardinals can behave like large cardinals. An old question of Magidor addressing these issues is: can we get principles like the tree property or ITP simultaneously for every regular cardinal greater than ω1? A positive answer would require many failures of SCH. In this paper we focus on ITP, the strongest of our key examples, and its relation to singular cardinal combinatorics. This is of particular interest because failure of SCH is an example of anticompactness, and so it is difficult to combine it with principles like ITP.

中文翻译：

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奇异基数假设的不可言喻的树属性和失败

集合论中一个长期存在的项目是分析在宇宙中可以获得多少紧凑性。紧凑性是一种现象，如果某个属性适用于一个对象的所有小子结构，那么它也适用于整个对象。特别感兴趣的紧凑性属性是遵循大基数的组合原则，但可以被迫在后继者中保持。关键示例包括（按强度增加的顺序）方块的失败、树属性和不可言喻的树属性 (ITP)。这些原则“捕捉”了某些大红雀的组合本质。在不可访问的基数上，树属性相当于弱紧性；ITP 相当于超紧凑性。将这些原则强加给继任者可以告诉我们小红雀在多大程度上可以像大红雀一样行事。Magidor 解决这些问题的一个老问题是：对于每个大于 ω1 的常规基数，我们能否同时获得树属性或 ITP 等原则？肯定的答案需要 SCH 多次失败。在本文中，我们重点讨论 ITP，我们最重要的关键示例，以及它与奇异基数组合的关系。这是特别有趣的，因为 SCH 的失败是反紧凑性的一个例子，因此很难将其与 ITP 等原则结合起来。