Talk:Boolean-valued model

metamathematics

I've removed this passage on metamathematics for now and reproduce it here for reference. Something like this needs to be discussed later, when the article describes how to use BVMs to get independence results, but it'll need some work at that time. (For one thing, the term "set theory" ought not to be used synonymously with ZFC; the latter is a partial axiomatization of set theory, not set theory itself). --Trovatore 17:50, 13 November 2005 (UTC)

In order to construct a Boolean valued model of set theory, we needed to start with a model V of set theory. At first sight this seems to be a problem, as we cannot prove the existence of such a model in set theory (unless set theory turns out to be inconsistent!) In practice this does not matter, because the theory of Boolean valued models is used mainly to prove relative consistency results. For example, Cohen did not prove that the continuum hypothesis is unprovable in set theory: what he proved was that if set theory is consistent then so is set theory with the negation of the continuum hypothesis, and for this it is enough to prove that if there exists a model V of set theory, then there exists a model in which the continuum hypothesis is false.

Relationship to forcing

This section is getting unwieldy, trying to explain it from many different points of view (by the way, leaving out the most usual way that set theorists get the Boolean algebra from the poset, which is to use the algebra of regular open subsets of the poset, the topology being generated by cones). Worse, it doesn't make the connection with syntactic forcing.

The big advantage of BVMs is that you can force over V, having an actual semantics rather than just defining the forcing relation syntactically, and you don't have to talk about generic objects that "don't exist". The disadvantages are that the semantics are not two-valued, and the combinatorics are usually harder to work with than in the underlying poset. These are the main points that need to be made in this section; all the stuff about generic ultrafilters and BA homomorphisms is interesting but of secondary importance. I'm not sure how best to fix it. I don't want to just cut out big chunks of good, interesting stuff, but maybe it could be moved to a different section, or made a subsection. I'll take a crack at it later today. --Trovatore 16:44, 14 November 2005 (UTC)

Maybe we should have two sections: keep the present one as a low level explanation for non set theorists, and add a more advanced section with the extra topics you mention. I didnt give the construction of the complete Boolean algebra via regular open sets because I havnt yet thought of a clear way to explain it, but I agree that something about it should be added. R.e.b. 17:06, 14 November 2005 (UTC)

I saw your description as more technical! All in one's perspective, I suppose. Anyway I have a reasonable first cut in place; see what you think. --Trovatore 17:43, 14 November 2005 (UTC)

Definition

I moved this text:

Text needs to be written explaining additional restrictions on interpretation of equality, guaranteeing that it's an equivalence relation and that relations respect substitution of equal things

from the article, where it does _not_ belong. Added sec-stub to note that the definition is incomplete.

Possible error

"If g is a homomorphism from a Boolean algebra B to a Boolean algebra C and MB is any B-valued model of ZF (or of any other theory for that matter) we can turn MB into a C -valued model by applying the homomorphism g to the value of all formulas."

This is wrong. g has to be a homomorphism of complete BA, not just BA. Otherwise, the value of formulas with quantifiers will get messed up. The section says that moding out with any ultrafilter will give a model on the equivalence classes. In reality, only generic ultrafilters work. Am I right on this? DS — Preceding unsigned comment added by 132.64.72.24 (talk) 13:01, 12 December 2012 (UTC)