[FOM] CH and forcing

[Timothy Y. Chow]

Bob Lubarsky wrote:

> If a Boolean-valued model is reduced by a non-generic ultrafilter, how 
> does this get you anything coherent? I thought the point of genericity 
> is that if an existential statement is forced then so is a witness. If 
> you don't have that then the truth lemma doesn't go through, and you end 
> up having no control over the resulting structure. I don't doubt that 
> V-check looks reasonable, but I'd think the new universe wouldn't even 
> provably model ZF.

The new universe does model ZF; the machinery of Boolean-valued models and 
ultrafilters is enough to ensure this.  What genericity ensures is a 
well-founded model that contains the ultrafilter that you're reducing by.  
This is explained in Bell's book on Boolean-valued models.

Tim