[FOM] CH and forcing
ablass at umich.edu
Tue Jul 5 13:14:55 EDT 2011
I agree that it's not easy to find in the literature the fact that you can get two-valued but not-necessarily-well-founded models by first forming the usual Boolean-valued model and then dividing by an arbitrary (not necessarily generic) ultrafilter. It was, however, published quite early by Vopenka; I believe it's in his very first papers about what he called the nabla model of set theory. There's a very systematic treatment of this and many related topics in the book "Theory of Semisets" by Vopenka and Hajek. Unfortunately, that book is not designed for browsing; you pretty much have to start from the beginning and read straight through in order to assimilate all the notational conventions.
Perhaps I should add a word of caution about the models obtained in this way: Not only are they usually not well-founded, they're usually not even omega-models. The "ground model" inside such a forcing extension (i.e., the extension of the predicate often written "V-check") is not isomorphic to the model you began with but rather to an elementary extension of it (specifically, a Boolean ultrapower). In principle, this is no problem, but in practice it's easy to get confused, especially if one is accustomed to the usual picture, where one uses a generic ultrafilter and gets a copy of the original model as V-check in the extension.
On Jul 4, 2011, at 11:57 AM, Aatu Koskensilta wrote:
> Quoting Andreas Blass <ablass at umich.edu>:
>> If one insists on two-valued models, the assertion is still correct, provided one does not require models to be well-founded.
> It's standard in expositions of forcing to restrict one's attention to well-founded models. Since all the necessary inductions and recursions involve only definable predicates this is clearly an overkill; all we need is that the ground model satisfied these and those induction and recursion principles (e.g. regularity given sufficient amount of replacement or whatnot). And, as those in the know very well know, we can force over e.g. uncountable models and obtain (sometimes necessarily) ill-founded models, just by boneheadedly going through the formal motions.
> Is there in the literature any systematic, clear, motivated, account of forcing covering all this? My personal experience is that everyone knows these things, but I'd be hard pressed if someone asked for reference.
> Aatu Koskensilta (aatu.koskensilta at uta.fi)
> "Wovon man nicht sprechen kann, darüber muss man schweigen"
> - Ludwig Wittgenstein, Tractatus Logico-Philosophicus
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