[FOM] CH and forcing

[Andreas Blass]

Roger Bishop Jones asked about the assertion

> "Every model of ZFC has a forcing extension in which CH
> fails."
>
In this context, I would interpret "forcing extension" to mean a 
Boolean-valued model.  With this interpretation, the assertion is correct.

If one insists on two-valued models, the assertion is still correct, 
provided one does not require models to be well-founded.  One can get a 
two-valued model as the quotient of a Boolean-valued model by any 
ultrafilter (not necessarily generic).

If, however, one wants two-valued, well-founded (or, equivalently, 
transitive) models, then there is indeed a problem when the given model is 
uncountable.

Andreas Blass