HOD中的紧性现象与Magidor覆盖定理的最优性
Compactness phenomena in HOD and the Optimality of Magidor's Covering theorem
AI总结：
该研究围绕集合论全域V与HOD的紧性现象，证明了奇异强极限基数的幂集分歧特性，回答了Hayut的问题，并揭示Magidor覆盖定理的最优性。
AI中文摘要：
我们延续了Goldberg与Poveda（文献[GolPov]）开创的、关于集合论全域V与HOD之间紧性现象的研究，重点关注V与HOD的幂集函数相关的紧性现象。我们证明了：（1）具有不可数共尾性的奇异强极限基数，不可能是$\boldsymbol{\textit{P}}(\boldsymbol{\textit{·}})$与$\boldsymbol{\textit{P}}^{\text{HOD}}(\boldsymbol{\textit{·}})$首次出现分歧的位置；（2）假设存在一个可测基数，$\boldsymbol{\textit{ℵ}}_\boldsymbol{\textit{ω}}$可以成为$\boldsymbol{\textit{P}}(\boldsymbol{\textit{ℵ}}_\boldsymbol{\textit{ω}})≠\boldsymbol{\textit{P}}^{\text{HOD}}(\boldsymbol{\textit{ℵ}}_\boldsymbol{\textit{ω}})$的首次分歧位置，这回答了Hayut提出的问题；（3）若$\boldsymbol{\textit{κ}}$是具有不可数共尾性的奇异强极限基数，且HOD对小于等于$\boldsymbol{\textit{κ}}^+$的基数是正确的，同时HOD在$\boldsymbol{\textit{κ}}^+$之下成立广义连续统假设（GCH），则$(\text{HOD}, V)$具有$\text{cf}(\boldsymbol{\textit{κ}})^+$-覆盖性质。我们还证明了（3）中的GCH假设是必要的，这表明Magidor经典覆盖定理是最优的。
英文摘要：
We continue the study of compactness phenomena between the set-theoretic universe and $\mathrm{HOD}$ initiated by Goldberg--Poveda \cite{GolPov}. We focus on compactness phenomena around the power-set functions of $V$ and $\mathrm{HOD}$. We prove: (1) A singular strong limit cardinal with uncountable cofinality cannot be the first place where $\mathcal{P}(\cdot )$ and $ \mathcal{P}^{\mathrm{HOD}}(\cdot)$ disagree. (2) Assuming the existence of a measurable cardinal, $\aleph_ω$ can be the first place where $\mathcal{P}(\aleph_ω)\neq \mathcal{P}^{\mathrm{HOD}}(\aleph_ω)$, answering a question of Hayut. (3) If $κ$ is strong limit singular of uncountable cofinality, $\mathrm{HOD}$ is correct about cardinals less than or equal to $κ^+$ and the GCH holds in $\mathrm{HOD}$ below $κ^+$ then $(\mathrm{HOD}, V)$ has the $\mathrm{cf}(κ)^+$-cover property. We also show that the GCH assumption in (3) is necessary, which demonstrates that Magidor's classical Covering Theorem is optimal.
