This seems like a natural question to ask, but I've not seen it discussed in my reading around (limited to Easton's paper, the third edition of Jech's *Set theory* and a small handful of articles). What *do* you get when forcing with a proper class of conditions that you don't get when forcing with, say, a set of conditions larger than one's model of set theory? Or rather, why isn't a large set of conditions enough?

Blass makes a throwaway remark in his 1984 paper *The interaction between category theory and set theory*

Although this approach [reflection principles] was first proposed in connection with the problem of foundations for category theory, it is natural to use it whenever objects seem to be too large to be coded as sets. In particular, it seems to me that it should be of some use in clarifying forcing with proper classes by making the natural (regular open) Boolean algebra available even though it is superlarge.

It seems to indicate that if we accept some sort of reflection principle, use an innaccessible cardinal $\kappa$ (or similar) - hence a Grothendieck universe - and a set of forcing conditions larger than $\kappa$, then we should arrive at our goal without using a proper class of conditions.

Alternatively, cannot one (ok, this is very naive, but this is why I'm asking) consider an inaccessible in ZFC and thus cook up a model of NBG, and then work with that a la Easton - and then at the end turn around a say 'Ahah! I was working in ZFC the whole time!'

One reason I ask is that in the paper *Injectivity, projectivity, and the axiom of choice*, Blass gives a symmetric model of ZFA with no nontrivial injective abelian groups using an uncountable set of atoms and a base model of ZFCA whose sets were in some sense 'small' (they arise, if I understand correctly, using the cumulative hierarchy generated from $A$ in the usual sense, but only taking *countable* subsets of $A$ at the first stage, rather than all of $\mathscr{P}A$). However, he gives a model of ZF with no nontrivial injective abelian groups using forcing involving a proper class of conditions. (Notice that Jech-Sochor is not useful in its usual statement because a global statement about a proper class of objects is required.) Perhaps the techniques given in Blass' Theorem 3.2 have been given a general treatment by now, I do not know.