Generalizing Hartogs' Trichotomy Theorem

选择公理（AC）与二阶Henkin逻辑中三分律的独立性

AC and the Independence of the Law of Trichotomy in Second-Order Henkin Logic

本文研究带Henkin解释的二阶谓词逻辑中的1-1型Ackermann选择公理集HAC，限定相关模型证明的适用范围，并证明Hartogs版本三分律独立于HAC乃至所有Ackermann公理，最后提出开放问题。

An Extension of Shelah's Trichotomy Theorem

In Sh506, Shelah develops the theory of $\mathrm{pcf}_I(A)$ without the assumption that $|A|<\min (A)$, going so far as to get generators for every $λ\in\mathrm{pcf}_I(A)$ under some assumptions on $I$. Our main theorem is that we can also generalize Shelah's trichotomy theorem to the same setting. Using this, we present a different proof of the existence of generators for $\mathrm{pcf}_I(A)$ which is more in line with the modern exposition. Finally, we discuss some obstacles to further generalizing the classical theory.