# Re: Paper and slides on indefiniteness of CH

Dear Bob,

Here is a precise statement. Suppose every set $X$ belongs to an inner model of a measurable Woodin (above X).

Suppose there is a club class $C$ of cardinals such that for all $\gamma \in C$,

$(V_{\gamma},C\cap \gamma) \not\vDash \textsf{ZFC}$

Then there is an inner model $N$ of $V$ such that $N \vDash \textsf{ZFC + GCH}$ and such that $V$ is a class-generic extension of $N$.

(Nothing special about GCH here).  Though here one can require that $N$ is a fine-structure model if one wants.

The class $C$ can always be forced without adding sets.

If one has inner model theory for measurable Woodin cardinals then the class $C$ is unnecessary.

Regards,
Hugh