发表机构
Universität Hamburg; CUNEF Universidad（汉堡大学; CUNEF大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文研究集合论强制对exacting与ultraexacting基数的影响，证明I2嵌入产生具有真类exacting基数的ZFC模型，并探讨HOD猜想下相关性质及强制扩张。
AI 中文摘要
受Aguilera--Bagaria--Lücke以及Aguilera--Bagaria--Goldberg--Lücke（ABGL）近期工作的启发，我们分析了集合论强制对exacting基数与ultraexacting基数类别的各种影响。利用Prikry强制的Magidor支撑积，我们证明了一个I2嵌入产生一个传递的、集合大小的ZFC模型，其中包含一个真类大小的exacting基数。我们证明，在Woodin的HOD猜想下，不存在一个exacting基数是exacting基数的极限。相反，我们证明了包含超越选择的大基数的ZF模型具有强制扩张，这些扩张是ZFC的模型，在其中一个正则基数是ultraexacting基数的平稳极限。我们还证明，假设HOD猜想，经典的Levy--Solovay定理的一个类比对于exacting和ultraexacting基数成立。最后，我们分析了exacting基数在HOD中所具有的大基数性质。我们证明，如果λ是exacting的且V_λ满足HOD假设，则λ在HOD中具有强的大基数性质。延续ABGL中的强制构造，我们从一个具有I2嵌入的ZFC模型出发，产生一个ZF模型，在其中exacting基数λ的后继基数在所有形如HOD_x（其中x⊆λ）的模型中都是extendible的。
英文摘要
Motivated by recent work of Aguilera--Bagaria--Lücke and Aguilera--Bagaria--Goldberg--Lücke (ABGL), we analyze various effects of set-theoretic forcing upon the classes of exacting and ultraexacting cardinals. Using Magidor support products of Prikry forcings, we prove that an I2-embedding yields a transitive, set-sized model of ZFC with a proper class of exacting cardinals. We show that under Woodin's HOD Conjecture, no exacting cardinal can be a limit of exacting cardinals. In contrast, we prove that models of ZF containing large cardinals beyond choice have forcing extensions that are models of ZFC in which a regular cardinal is a stationary limit of ultraexacting cardinals. We also show that, assuming the HOD Conjecture, an analogue of the classical Levy--Solovay theorem holds for exacting and ultraexacting cardinals. Finally, we analyze the large cardinal properties possessed by exacting cardinals in HOD. We prove that if $λ$ is exacting and $V_λ$ satisfies the HOD Hypothesis, then $λ$ has strong large cardinal properties in HOD. Carrying forward forcing constructions from ABGL, we start with a model of ZFC with an I2-embedding and produce a model of ZF in which the successor of an exacting cardinal $λ$ is extendible in all models of the form $\mathrm{HOD}_x$ for $x\subseteq λ$.
