[FOM] CH and forcing
Timothy Y. Chow
tchow at alum.mit.edu
Sat Jul 9 23:11:28 EDT 2011
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.
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