For an inaccessible cardinal κ , the super tree property (ITP) at κ holds if and only if κ is supercomact. However, just like the tree property, it can hold at successor cardinals. We show that ITP holds at the successor of the limit of ω many supercompact cardinals. Then we show that it can consistently hold at ℵ ω +1 . We also consider a stronger principle, ISP and certain weaker variations of it. We determine which level of ISP can hold at a successor of a singular. These results fit in the broad program of testing how much compactness can exist in the universe, and obtaining large cardinal-type properties at smaller cardinals.

中文翻译：

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单数继承者的超级树属性

对于不可访问的基数 κ ，当且仅当 κ 是超紧凑时，κ 处的超级树属性（ITP）成立。但是，就像树属性一样，它可以保留在后继基数上。我们证明了 ITP 在 ω 许多超紧基数的极限的后继处成立。然后我们证明它可以始终保持在 ℵ ω +1 。我们还考虑了更强的原则、ISP 及其某些较弱的变体。我们确定哪个级别的 ISP 可以容纳单数的后继者。这些结果适用于测试宇宙中可以存在多少紧凑性的广泛程序，并在较小的基数上获得大的基数类型的属性。