This is a talk in the 2021 Boise Extravaganza in Set Theory on 2021 June 19.

[slides]

The mantle, the intersection of all grounds, is the largest set forcing-invariant inner model. Fuchs, Hamkins, and Reitz proved that the mantle is not absolute—it is consistent that the mantle of the mantle is a yet thinner inner model. As such, the sequence of inner mantles, obtained by transfinitely iterating taking the mantle of the mantle of…, consistently does not stabilize after just one step. Reitz and myself showed that for any ordinal $\eta$ it is consistent for this sequence to stabilize at length exactly $\eta$. Left open from that work was the question whether it’s possible for the sequence to break down at limit stages. In this talk I will discuss how to use tree iterations to code sets into inner mantles. As an application, we will see that (1) it is consistent that the $\omega$-th inner mantle is an undefinable class and (2) it is consistent that the $\omega$-th inner mantle is a model of ZF plus the failure of AC.