On the optimality of the HOD dichotomy

TL;DR

This paper investigates the properties of HOD in relation to large cardinals, establishing consistency results and exploring the extent of weak covering failures under the HOD hypothesis.

Contribution

It provides new consistency results about the relationship between HOD and large cardinals, including extendibility and supercompactness, and addresses questions about weak covering failures.

Findings

First extendible cardinal can be the first strongly compact in HOD.

Under HOD hypothesis, the first extendible is C^{(1)}-supercompact in HOD.

There are many singulars where HOD and V agree on cofinality and successor cardinals.

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 $\delta$ 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 $\kappa<\delta$ where $\cf^{\HOD}(\kappa)=\cf(\kappa)$ and $\kappa^{+\HOD}=\kappa^{+}.$ In contrast, we also show that the $\HOD$ hypothesis is consistent with $\delta$ carrying a club of $\HOD$-regulars cardinals $\kappa$ such that $\kappa^{+\HOD}<\kappa^{+}$. Finally, we close the manuscript with a discussion about the $\HOD$ hypothesis and $\omega$-strong measurability.