Collapsing the cardinals of $HOD$

TL;DR

This paper constructs a model where the HOD (Hereditarily Ordinal Definable sets) has smaller cardinals than the ambient universe, demonstrating a collapse of cardinals within HOD under certain large cardinal assumptions.

Contribution

It introduces a method to create a model where all infinite cardinals in HOD are strictly less than in the universe, under GCH and supercompactness assumptions.

Findings

In the constructed model, $(eta^+)^{HOD} < eta^+$ for all infinite $eta< ext{cardinal}$.

The model preserves strong inaccessibility of $ ext{kappa}$ while collapsing cardinals in HOD.

The rank-initial segment $W_ ext{kappa}$ satisfies ZFC with the same cardinal collapsing property.

Abstract

Assuming that $GCH$ holds and $\kappa$ is $\kappa^{+3}$-supercompact, we construct a generic extension $W$ of $V$ in which $\kappa$ remains strongly inaccessible and $(\alpha^+)^{HOD} < \alpha^+$ for every infinite cardinal $\alpha < \kappa$. In particular the rank-initial segment $W_\kappa$ is a model of ZFC in which $(\alpha^+)^{HOD} < \alpha^+$ for every infinite cardinal $\alpha$.