This is joint with Jonas Reitz

[PDF] [arXiv]

Abstract We present a class forcing notion $\mathbb M(\eta)$, uniformly definable for ordinals $\eta$, which forces the ground model to be the $\eta$-th inner mantle of the extension, in which the sequence of inner mantles has length at least $\eta$. This answers a conjecture of Fuchs, Hamkins, and Reitz [FHR15] in the positive. We also show that $\mathbb M(\eta)$ forces the ground model to be the $\eta$-th iterated $\mathrm{HOD}$ of the extension, where the sequence of iterated $\mathrm{HOD}$s has length at least $\eta$. We conclude by showing that the lengths of the sequences of inner mantles and of iterated $\mathrm{HOD}$s can be separated to be any two ordinals you please.