[FOM] CH and forcing

[Roger Bishop Jones]

I have twice recently read the following assertion:

"Every model of ZFC has a forcing extension in which CH 
fails."

Once by Woodin, and once by Hamkins.

"Every model" is clearly too strong here, since not every 
model has any extensions at all (and the truth value of CH 
in second order ZFC is not known).

"Every countable transitive model" is more like the result 
usually proven, and the proof as I understand it does depend 
on the model being countable.

Is there a stronger result?

Roger Jones