TITLE: Class forcing: Pelletier vs Friedman
QUESTION [8 upvotes]: [Apologies in advance for a fluffy question]
I'm reading this old paper by Pelletier, where he gives a Boolean-valued model version of class forcing, assuming that the Boolean algebra in question can be written as an $Ord$-indexed increasing union of CBAs $B_\alpha$. The model is given by induction, as usual, with a twist:

$^RV^B_0 = \emptyset$
$^RV^B_{\alpha+1} = \lbrace u \in B_{\alpha+1}^{dom(u)} |\ dom(u) \subseteq {}^RV^B_\alpha\rbrace$
$^RV^B_\lambda = \bigcup_{\alpha \lt \lambda} {}^RV^B_\alpha$ for $\lambda$ limit.
$^RV^B = \bigcup_{\alpha \in Ord} {}^RV^B_\alpha$

So in a sense, our hierarchy is restricted as to what can appear at the $\alpha$th stage. Given a certain condition on $B$ called ARP (I can supply details if desired), he shows that $^RV$ is a model of ZFC-Powerset. Another condition on $B$ gives us Powerset, and all this works in the usual Easton situation (ARP seems to be related to Easton support, but I may be wrong).
On the other hand, Friedman (although I'm reading his book), given a proper class $P$ of conditions defines names and interpretations as per usual for an ordinary forcing with no restrictions to get $M[G]$ (here $M$ satisfies a relativised $V=L$). He then shows that with a pretameness condition on P - that every set-indexed collection of dense (definable-with-parameters) classes in $P$, at the cost of passing to a $q\leq p$, there are subsets of each class whose down-closure is dense below $q$ - we have that $M[G]$ is a model of ZF(C)-Powerset. Then with an additional tameness condition on $P$, which refers to the  cumulative hierarchy of the base model $M$, he shows $M[G]$ is a model of ZF(C).
What I want to know is how these two approaches can get away with introducing the stratification in the two different parts of constructing the new, class-forced model. Pelletier uses ARP to show that $|| - ||$ is well-defined (so not having to quantify over classes) using his stratification of $B$, then the other condition to show that powerclasses are sets. Friedman uses pretameness to show $\Vdash$ is definable and then proves the relevant axioms hold, then uses tameness, and its condition using $V_\alpha^M$ to get powersets.
Perhaps it is just the different approaches of BVM and forcing using conditions, but I'm trying to take a third approach, and seeing where the stratification restriction is used seems to be crucial for what I need.
EDIT:  Then again, Jech just says to form the BVM $V^{B_\alpha}$ for each $\alpha$ and then take the union of them all, so this stratifies in yet another way.

REPLY [4 votes]: I think the last paragraph of Joel Hamkins's answer provides a useful way to think about these sorts of forcing.  Imagine that you have an inaccessible $\kappa$ (or, perhaps better, a $\kappa$ such that $V_\kappa$ is an elementary submodel of the universe) and that you're forcing with a poset $\mathbb P\subseteq V_\kappa$.  You get a forcing extension of the universe (since $\mathbb P$ is a mere set in this picture), but your objective is to get a ZFC model by restricting this model to ranks $<\kappa$.  Several issues arise.  You need to decide what "restricting to ranks $<\kappa$" means.  It could mean taking $V_\kappa$ in the forcing extension.  It could mean taking those elements of the forcing extension that have names (in the ground model) of ranks $<\kappa$.  I've seen both versions, and I don't think they're equivalent.
If you want to use Boolean-valued models, then the complete Boolean algebra $\mathbb B$ associated to $\mathbb P$ is one level higher in the cumulative hierarchy than $\mathbb P$, so you need to reconsider anything involving ranks of names.
These issues are "formal" in the sense that they concern how you want to set up the machinery.  But there are also some substantive issues that you must face no matter how you set up the machinery.  For example, as Joel mentioned, you'd better make sure you haven't added $\kappa$ new reals, as they would ruin ZFC restricted to ranks $<\kappa$.  (Whether they ruin the power set axiom or the replacement scheme seems to depend on how you choose to "restrict to ranks $<\kappa$".)