On the optimality of the HOD dichotomy
Gabriel Goldberg, Jonathan Osinski, Alejandro Poveda
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 $δ$ 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 $κ<δ$ where $\cf^{\HOD}(κ)=\cf(κ)$ and $κ^{+\HOD}=κ^{+}.$ In contrast, we also show that the $\HOD$ hypothesis is consistent with $δ$ carrying a club of $\HOD$-regulars cardinals $κ$ such that $κ^{+\HOD}<κ^{+}$. Finally, we close the manuscript with a discussion about the $\HOD$ hypothesis and $ω$-strong measurability.