What is the least theory in which the following sentence is proved?
$ \exists M: M\text { is CTM(ZFC+ GCH)} \land \forall \kappa \in Card^M (\kappa > 1 \implies \\\exists N: N \text { is CTM(ZFC) } \land Card^N=Card^M \land \operatorname\neg \operatorname {CH}^N(\kappa))$
Where $Card^S=\{\kappa : S \models \exists x \, (\kappa=|x|)\}$
$\operatorname {CH}(\kappa) \iff \not \exists \lambda: \kappa < \lambda < 2^\kappa$
"$\operatorname {CH}^N..$" is the relativization of all quantifiers in the open expansion of "$\operatorname {CH}..$" by "$\in N$".
In English: There exists a countable transitive model $M$ of ZFC +GCH such that for every $M$-cardinal $\kappa$ (bigger than 1) there is a countable transitive model $N$ of ZFC such that the set of all $N$-cardinals is the set of all $M$-cardinals (so existence of injections between cardinals in both models is preserved), and such that $\kappa$ violate the continuum hypothesis in $N$.
