# Inner mantles and iterated HOD

This is a contributed talk at the 2019 ASL Annual North American Meeting on Tuesday, 21 May 2019. It is about joint work with Jonas Reitz.

An inner model $W \subseteq V$ is a *ground* if $V$ is a forcing extension of $W$ via a forcing notion $\mathbb{P} \in W$. The intersection of all the grounds is called the *mantle*, and it is the largest set-forcing-invariant inner model. Fuchs, Hamkins, and Reitz showed that there is a class forcing notion which forces the ground model to be the mantle of the extension. In that same article, they defined the sequence of *inner mantles*, obtained by iterated the definition of the mantle to get the mantle of the mantle, the mantle of the mantle of the mantle, and so on, and conjectured that there is a class forcing to force the ground model to be the $\eta$-th inner mantle of the extension for any ordinal $\eta$.

We answer their conjecture in the positive. 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 $\eta$. 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 $\eta$. Finally, we show 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.