2 problems

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…