Collapsing the cardinals of $HOD$

James Cummings, Sy David Friedman, Mohammad Golshani

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…