Large cardinals need not be large in HOD

TL;DR

This paper demonstrates that large cardinal properties in the universe do not necessarily reflect in HOD, showing that large cardinals can lose their large cardinal features when viewed inside HOD.

Contribution

It establishes that large cardinals like supercompact, measurable, and strong can fail to exhibit their properties in HOD, revealing limitations of their reflection.

Findings

Supercompact cardinals may not be weakly compact in HOD

There can be many supercompact cardinals in V with none in HOD

Similar results hold for measurable and strong cardinals

Abstract

We prove that large cardinals need not generally exhibit their large cardinal nature in HOD. For example, a supercompact cardinal $\kappa$ need not be weakly compact in HOD, and there can be a proper class of supercompact cardinals in $V$, none of them weakly compact in HOD, with no supercompact cardinals in HOD. Similar results hold for many other types of large cardinals, such as measurable and strong cardinals.