[FOM] CH and forcing

[lubarsky.robert at comcast.net]

Andreas Blass wrote: 

> Nevertheless, even 
with a non-generic ultrafilter, all the finitary, first-order consequences 
of "exact copy" remain correct; the reduction of V-check modulo any 
ultrafilter is an elementary extension of the ground model. 


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.


Bob Lubarsky