2 problems

Let $M$ be an inner model of $V$. Write $2^{\omega_1} \cap M$ for the subsets of $\omega_1$ belonging to $M$, and write $\operatorname{Ord}^{\omega_1} \cap M$ for the functions fro…

Let $alef_\omega$ be a strong limit cardinal. Write ${\sf TP}(\aleph_{\omega+1})$ for the tree property at its successor, ${\sf SCH}$ for the singular cardinal hypothesis, and…