Elementary chains in forcing extensions of $M_1$

Question

Let $M_1$ be the canonical inner model with one Woodin cardinal $\delta$. Now suppose that $\mathbb{P}$ is a forcing notion of size $< \delta$, which preserves $\omega_1$ and that $G$ is a generic filter for the forcing. Note that, assuming $M_1^{\sharp}$ exists, there exists a formula which defines $M_1$ in the generic extension properly. Suppose that in $M_1[G]$ the $(M_1)$-initial segment $(M_1)_{\alpha}$ has size $\aleph_1$.

My question is whether it is possible, for unboundedly many $\alpha < \aleph_2^{M_1[G]}$ to find an elementary continuous chain $(N_i \, : \, i \in \omega_1)$ of elementary submodels of $(M_1)_{\alpha}$ such that $\bigcup_{i < \omega_1} N_i = (M_1)_{\alpha}$, and such that for every model $N_i$ in the chain, its transitive collapse is an initial segment of $M_1$ again?

Note that the statement is true if $\mathbb{P}$ is just the trivial forcing, however I would be interested in the case when we collapse cardinals down to $\omega_1$.

Answer 1

i think the answer is yes.

let $M=M_1$. let $\lambda=\omega_2^M$ and say $G$ collapses $\omega_2$ to $\omega_1$. let $f\colon\omega_1\to\lambda$ be an increasing cofinal function. let construct a continuous chain $(M_i: i<\omega_1)$ of submodels of $M|\lambda$ such that $f(i)$ is in $M_i$.

let $N_i$ be the transitive collapse of $M_i$. then $N_i$ is an initial segment of $M$, this follows from comparison.

let then $M_{\omega_1}$ be the direct limit of $N_i$. there is a map $\pi\colon M_{\omega_1}\to M_i|\lambda$. this map has the range of $f$ in its range and its critical point is bigger than $\omega_1$, so it is identity.

you can do this with any $\gamma<\omega_3^M$. fix $f\colon\omega_1\to\lambda$ and $g\colon\omega_1\to\gamma$ that are increasing and cofinal. then construct a chain that exhaust $f$ and $g$. similar argument would show that $M_{\omega_1}$ is $M|\gamma$.