A Special Pair of Models for ZFC (New Version) 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. 
 A: I add a few more remarks to Prof. Caicedo's nice answer:
Suppose $V \subset W$, $\aleph_{1}^W=\aleph_{1}^V$ and $\aleph_2^W=\aleph_{\omega+1}^V$. Then
1) In $V,$ SCH holds at $\aleph_\omega$,
2) There is a $V$-stationary subset $\aleph_{\omega+1}^V$ which is not $W$-stationary,
3) There exists in $W$ an $\omega$-sequence of elements of $\aleph_\omega^V$ which is not
in $V$.
4) $W$ is not a forcing extension of $V$ by a proper forcing notion.
For more information see Cummings paper "Two Results in Combinatorial Set Theory".
A: 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.)
A: As Péter indicates, this is open. It is related to a question that is a favorite of many. Shelah has conjectured that if $M\subseteq N$ are (proper class transitive) models of $\mathsf{ZFC}$ and $\kappa$ is a cardinal in $M$ with succesor $\lambda$, and $\lambda$ is a cardinal in $N$, then in $N$ we have that $\mathrm{cf}(|\kappa|)=\mathrm{cf}(\kappa)$. Note that if there are two models as you inquire, then $\kappa=\aleph_\omega^M$ contradicts Shelah's conjecture. 
Shelah himself proved his conjecture in some cases, namely when, in $M$, $\kappa$ is either regular, or a singular such that $\square_\kappa$ holds. In fact, his proof naturally splits into two parts: 


*

*Under either assumption, in $M$, the combinatorial principle that we now call $ADS_\kappa$ holds. 

*If $M\subseteq N$, $(\kappa^+)^M$ is a cardinal of $N$, and $ADS_\kappa$ holds in $M$, then in $N$, $\mathrm{cf}(|\kappa|)=\mathrm{cf}(\kappa)$.


Here, $ADS_\kappa$ is the statement that there is an Almost Disjoint sequence of Sets, in the following sense: A sequences $(A_\alpha\mid\alpha<\kappa^+)$ such that each $A_\alpha$ is an unbounded subset of $\kappa$, and there are functions $g_\beta:\beta\to\kappa$ for all $\beta<\kappa^+$ such that for each such $\beta$, we have that $(A_\alpha\setminus g_\beta(\alpha)\mid\alpha<\beta)$ is a sequences of pairwise disjoint sets.
This appears in the Cardinal arithmetic book, Lemma 4.9 in Chapter VII (page 304), where Shelah remarks that we can also assume instead of 1. that, in $M$, $\kappa$ is singular and $\mathrm{pp}(\kappa)>\kappa^+$. 
James Cummings, and later Cummings-Foreman-Magidor have studied Shelah's conjecture. The key references are 

James Cummings. Collapsing successors of singulars, Proc. Amer. Math. Soc., 125 (9), (1997), 2703–2709. MR1416080 (97j:03091),

and 

James Cummings, Matthew Foreman, and Menachem Magidor. Squares, scales and stationary reflection, J. Math. Log., 1 (1), (2001), 35–98. MR1838355 (2003a:03068).

Cummings-Foreman-Magidor prove that rather than square, assuming the weak square principle $\square^*_\kappa$ or the existence of a very good scale suffice. Cummings shows that there are many difficulties in obtaining models contradicting Shelah's conjecture. He also indicates a specific way that one could violate the conjecture and in fact reach your setting, namely, if one has that $$ (\aleph_{\omega+1},\aleph_\omega)\twoheadrightarrow(\aleph_2,\aleph_1) $$ and there is a Woodin cardinal $\delta$. The specific instance of Chang's conjecture needed here is open. It implies that $\{X\subseteq\aleph_{\omega+1}\mid \mathrm{ot}(X)=\aleph_2\}$ is stationary, so we can force with the full stationary tower (for $\delta$) below this set, and $V,V[G]$ is a model of your situation.  
Additional later work by Cummings-Foreman-Magidor (on "canonical structure") indicates further difficulties. 
A: That's a famous unsolved problem. 
L. Bukovsky, E. Coplakova-Hartova: Minimal collapsing extension of models of ZFC, Annals of Pure and Applies Logic, 46(1990), 265-298. 0
