4 problems

Downwards Directedness Hypothesis. All grounds are downwards directed.

Let $\eta$ be an ordinal and let $V$ be a model of $\mathsf{ZFC}$. An inner mantle is obtained by iterating the operation that sends a model to the intersection of its grounds. Fuc…

Let $V$ be a model of ${\rm ZFC}$ and let $\alpha\leq\mathop{\rm ORD}$. In another model $\overline{V}$ of ${\rm ZFC}$, write $({\mathord{\rm M}}^\alpha)^{\overline{V}}$ and…

Let $V$ be a model of ${\rm ZFC}$, and let $\mathord{\rm M}^\alpha$ denote the $\alpha$th inner mantle obtained by starting with $V$, taking the mantle at successor stages, and int…