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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?

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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 $W=L[G]$.

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  • $\begingroup$ Thanks, it was unexpected for me. What if we assume there are no large cardinals in V? $\endgroup$
    – Mohammad Golshani
    Dec 9, 2013 at 4:32
  • $\begingroup$ I think your $W[G]$ is $V[G]$. $\endgroup$
    – Mohammad Golshani
    Dec 9, 2013 at 4:36
  • $\begingroup$ Thanks, I fixed the typo. Yes, that is the question to think about, what we can say without large cardinals. $\endgroup$
    – Andrés E. Caicedo
    Dec 9, 2013 at 5:35

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