# Intermediate submodels and the continuum hypothesis

Let $V$ be a model of $ZFC+GCH$ and let $V[G]$ be a generic extension of $V$ in which $CH$ fails.

Question 1. Is there a model $W$ such that:

1) $V \subseteq W \subseteq V[G],$

2) $W\models CH,$

3) $W$ and $V[G]$ have the same cardinals.

Question 2. The same question as above, this time assuming $V=L$ and there are no large cardinals?

-

The answer is no in general. If $V[G]$ is a model of Martin's maximum and $W$ is an inner model of $V[G]$ with the same $\omega_2$, then $\mathcal P(\omega_1)^W=\mathcal P(\omega_1)^{V[G]}$, so $\mathsf{CH}$ fails in $W$. In fact, many strong reflection principles can be used here instead of $\mathsf{MM}$, for instance we could have $V[G]$ a model of Rado's conjecture. Also, (if there is a $\Sigma_1$-reflecting cardinal in $L$) there is a forcing extension $L[G]$ of $L$ where $\mathsf{BPFA}$ holds, and if $W$ is an inner model of $L[G]$ with the same $\omega_2$, then $W=L[G]$.
I think your $W[G]$ is $V[G]$. –  Mohammad Golshani Dec 9 '13 at 4:36