The super tree property at the successor of a singular

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.

对于一个不可达基数κ，κ上的超树性质（ITP）成立当且仅当κ是超紧的。然而，就像树性质一样，它可以在后继基数上成立。我们证明ITP在ω多个超紧基数的极限的后继上成立。然后我们证明它可以一致地在ℵ₍ω + 1₎上成立。我们还考虑一个更强的原理，ISP以及它的某些更弱的变体。我们确定ISP的哪个层级可以在一个奇异基数的后继上成立。这些结果符合检验宇宙中可以存在多少紧致性以及在较小基数上获得大基数类型性质的广泛计划。