TITLE: A Special Pair of Models for ZFC (New Version)
QUESTION [7 upvotes]: Are there two models $M$ and $N$ for $\text{ZFC}$ such that:
(1) $M\subseteq N$
(2) $\aleph_{1}^{N}=\aleph_{1}^{M}$
(3) $\aleph_{2}^{N}=\aleph_{\omega +1}^{M}$
Update: According to Peter's useful answer it seems this problem is still open. So my question is about "partial" or "similar" results with different (but not too different) $\aleph$s. Please give me some references too.

REPLY [6 votes]: Regarding your question about the situation at other $\aleph$s: Lemma 3.1 from Cummings' paper "Collapsing successors of singulars," together with the fact (due to Shelah, I believe) that, if $\overrightarrow{f}$ is a scale of length $\aleph_{\omega+1}$, then there is a club $C$ in $\aleph_{\omega+1}$ such that every $\alpha \in C \cap \mathrm{cof}(\geq \aleph_4)$ is good for $\overrightarrow{f}$, yields that, if $5 \leq n < \omega$, then there cannot be inner models $V\subset W$ such that $\aleph_{\omega+1}^V = \aleph_n^W$.
Also, to add to the list of restrictions your desired situation would impose on $V$ and $W$ already given by Andres and Mohammad, it follows from results of Sharon and Viale from their paper "Some consequences of reflection on the approachability ideal" that if $V\subset W$ are inner models and $\aleph_{\omega+1}^V = \aleph_n^W$ for some $n<\omega$, then, in $W$, a form of simultaneous stationary reflection must fail at $\aleph_n^W$. More precisely, in $W$, for all stationary $S\subseteq \aleph_n^W$, there are $\{S_i \mid i<\omega \}$ such that each $S_i$ is a stationary subset of $S$ and there is no $\alpha < \aleph_n^W$ such that each $S_i$ reflects at $\alpha$. (Note that this is trivial if $n=1$, since no stationary subsets of $\aleph_1$ reflect.)