On the optimality of the HOD dichotomy

Abstract

In the first part of the manuscript, we establish several consistency results concerning Woodin's $\HOD$ hypothesis and large cardinals around the level of extendibility. First, we prove that the first extendible cardinal can be the first strongly compact in HOD. We extend a former result of Woodin by showing that under the HOD hypothesis the first extendible cardinal is $C^{(1)}$-supercompact in HOD. We also show that the first cardinal-correct extendible may not be extendible, thus answering a question by Gitman and Osinski \cite[\S9]{GitOsi}. In the second part of the manuscript, we discuss the extent to which weak covering can fail below the first supercompact cardinal $\hat{I}´$ in a context where the HOD hypothesis holds. Answering a question of Cummings et al. \cite{CumFriGol}, we show that under the $\HOD$ hypothesis there are many singulars $\hat{I}º<\hat{I}´$ where $\cf^{\HOD}(κ)=\cf(κ)$ and $κ^{+\HOD}=κ^{+}.$ In contrast, we also show that the $\HOD$ hypothesis is consistent with $\hat{I}´$ carrying a club of $\HOD$-regulars cardinals $\hat{I}º$ such that $κ^{+\HOD}<κ^{+}$. Finally, we close the manuscript with a discussion about the $\HOD$ hypothesis and $\ddot{I}‰$-strong measurability.