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 Special Pair of Models for ZFC (New Version)
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: 


1. Under either assumption, in $M$, the combinatorial principle that we now call $ADS\_\kappa$ holds.
2. 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*](http://www.ams.org/journals/proc/1997-125-09/S0002-9939-97-03995-6/S0002-9939-97-03995-6.pdf), Proc. Amer. Math. Soc., **125 (9)**, (1997), 2703–2709. [MR1416080 (97j:03091)](http://www.ams.org/mathscinet-getitem?mr=1416080),
> 
> 
> 


and 



> 
> James Cummings, Matthew Foreman, and Menachem Magidor. [*Squares, scales and stationary reflection*](http://www.math.cmu.edu/users/jcumming/papers/jpaper.pdf), J. Math. Log., **1 (1)**, (2001), 35–98. [MR1838355 (2003a:03068)](http://www.ams.org/mathscinet-getitem?mr=1838355).
> 
> 
> 


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. 


