We show that the consistency of the theories “ZF + ¬ACω + GCH holds below אω + There is an injection f : אω+2 → ℘(אω) + Both אω and APאω fail” and “ZF + ¬ACω + GCH holds below אω + There is an injection f : אω+2 → ℘(אω) + אω+1 satisfies the tree property” follow from the appropriate supercompactness hypotheses. These provide answers in a choiceless context to certain long-standing open questions concerning SCH, weak square, approachability, and the tree property. There is nothing special about אω+2, and the injection into ℘(אω) can be from any ordinal λ (which means that pcf theory can fail badly in the models constructed if λ ≥ אω4). ∗2010 Mathematics Subject Classifications: 03E25, 03E35, 03E45, 03E55. †

中文翻译：

---

关于无选择上下文中 SCH 的弱平方、可接近性、树属性和失败

我们证明了理论的一致性“ZF + ¬ACω + GCH 低于 אω + 存在注入 f : אω+2 → ℘(אω) + אω 和 APאω 都失败”和“ZF + ¬ACω + GCH 低于 אω” אω + 有一个注入 f : אω+2 → ℘(אω) + אω+1 满足树属性”遵循适当的超紧性假设。这些在无选择的上下文中为某些长期存在的开放性问题提供了答案，这些问题涉及 SCH、弱平方、可接近性和树属性。אω+2 没有什么特别之处，并且注入 ℘(אω) 可以来自任何序数 λ（这意味着如果 λ ≥ אω4，则 pcf 理论在构建的模型中可能会严重失败）。∗2010 数学学科分类：03E25、03E35、03E45、03E55。†