Compactness phenomena in HOD and the Optimality of Magidor's Covering theorem

Abstract

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, ${\aleph }_{\omega }$ can be the first place where $\mathcal{P}({\aleph }_{\omega })\ne {\mathcal{P}}^{\mathrm{HOD}}({\aleph }_{\omega })$, answering a question of Hayut. (3) If $\kappa$ is strong limit singular of uncountable cofinality, $\mathrm{HOD}$ is correct about cardinals less than or equal to ${\kappa }^{+}$ and the GCH holds in $\mathrm{HOD}$ below ${\kappa }^{+}$ then $(\mathrm{HOD},V)$ has the $\mathrm{cf}(\kappa {)}^{+}$-cover property. We also show that the GCH assumption in (3) is necessary, which demonstrates that Magidor's classical Covering Theorem is optimal.