Dear Bob,

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

Suppose there is a club class of cardinals such that for all ,

Then there is an inner model of such that and such that is a class-generic extension of .

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

The class can always be forced without adding sets.

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

Regards,

Hugh