[FOM] CH and forcing

[Andreas Blass]

In an earlier message, I 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.

In reply to this, Bob Lubarsky asked:

> 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 solution to this difficulty is a property of the usual construction of 
Boolean-valued models of ZFC often called the maximal principle: The 
Boolean truth value of formula of the form "(exists x) A(x)", although 
defined as the Boolean supremum of the truth values of instances A(p) 
where p ranges over all Boolean-valued names, is in fact equal to the 
truth value of A(p) for *one* suitably chosen name p.  That p (or, more 
precisely, its equivalence class modulo the ultrafilter) serves as the 
required witness for (exists x) A(x).  The proof of the maximal principle 
obtains the desired p by patching together partial witnesses p'.  That is, 
one starts with all the Boolean algebra elements b for which some p' 
satisfies A(p') with truth value at least b, one chooses a maximal set of 
pairwise disjoint such elements b, and one defines p to agree with value 
(at least b) with an appropriate p' for each b in that antichain.  (The 
axiom of choice is needed here, in order to choose the antichain and the 
appropriate p'.  I believe I was careful enough, in my earlier message, to 
always talk about ZFC, not just ZF.  Without choice, Bob's objection is 
entirely correct.)

Andreas Blass