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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.

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The idea behind the remark quoted in the question was that, in situations ordinarily treated with proper-class forcing (e.g., Easton's theorem), the work can be transcribed rather routinely into a Feferman-style set theory (ZFC plus a constant $\kappa$ for an ordinal and axioms saying, one formula at a time, that $V_\kappa$ is an elementary submodel of the universe $V$). Just do with $V_\kappa$ what you would otherwise have done with $V$. Where (for example) Easton got arbitrary cardinal exponentiation at all regular cardinals, you'd now get arbitrary cardinal exponentiation only at all regular cardinals below $\kappa$, but that's "morally" or "intuitively" the same (and gives the same relative consistency result) because of the elementarity of $V_\kappa$ in $V$. This "large set" approach allows you to work with the framework of Boolean-valued models rather than forcing, whereas a proper-class forcing would, in general, need super-classes (yet another level higher in the cumulative hierarchy) to do this.

In both frameworks, the real issue is not whether you work with large (i.e., $\kappa$-sized or bigger) sets or with proper classes but rather what additional conditions you impose on your forcing notions (or Boolean-valued models). As Nate pointed out, you need some conditions (in either framework) to make sure you get a model of ZFC. If you just go blindly ahead (in either forcing), you could, for example, add a proper class (respectively a $\kappa$-sized family) of Cohen reals, so that the continuum will no longer be a set in your forcing extension (of $V$, respectively $V_\kappa$). Or you might collapse all the cardinals (resp. all the cardinals below $\kappa$).

Of course, some people might want to sacrifice (part of) ZFC and work with such "strange" models. If I remember correctly, Rudy Rucker once (before he turned to science-fiction writing) proposed working in the theory obtained from ZFC by deleting the power set axiom and adding Martin's Axiom for arbitrarily large collections of dense sets (so the continuum has to be a proper class). But here again, it seems to me that it makes little difference which framework you use.

Also, I recall that Sy Friedman did some work on super-class forcing. I don't know any of the details, but I would expect that this too can be easily recast in terms of forcing over a Feferman-style $V_\kappa$.

Finally, let me mention that, if you force in the Feferman framework, you actually have two choices for what should be the generic extension of $V_\kappa$. One is to take the elements of rank below $\kappa$ in the generic extension of the full universe. The other is to take the denotations of names whose rank is below $\kappa$. The two seem to coincide in nice cases, but I don't see any reason for them to coincide in general. (The second is what corresponds to proper-class forcing over $V$.)

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Thanks, Andreas. I'll respond in a little bit to this. –  David Roberts Oct 22 '12 at 19:06
    
Ah, I see where my misconception might be. In terms of sheaf-style forcing we should be taking small sheaves - those sheaves which are small colimits of representables. This should (I think!) correspond to the names-below-rank-$\kappa$. But small sheaves seem to be the right thing on large sites in any case. –  David Roberts Oct 22 '12 at 22:56
    
Hi Andreas, do you have a reference that discusses your last paragraph? –  David Roberts Oct 24 '12 at 4:50
    
David, I'm not aware of any reference for the choice between two ways of cutting off the model at rank $\kappa$. A careful analysis of the situation might get unpleasantly complicated, because the option using the names will depend on exactly how one defines names. In the Boolean-valued approach, they are likely to have a higher rank than in the poset approach to forcing. Also, I'm not aware of any situations where one gets particularly nice or unexpected results by making the choice cleverly. So there seems to be too much work and not enough payoff in trying to work this out carefully. –  Andreas Blass Oct 24 '12 at 17:51
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There is some danger associated with going too far outside the ground model. One of the key aspects of forcing is that we can understand it from inside the ground model: the forcing relation $p \Vdash \phi$ is definable! Of course, it is not possible to define this relation without understanding what conditions and names are. Working with sets or proper classes (definable classes in ZFC or actual classes in NBG) ensures we always understand what conditions and names are and that the forcing relation is definable.

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I'm not sure how this answers my question. In case it wasn't clear, I'm wondering what extra forcing/consistency results we get using class forcing over large set forcing. –  David Roberts Oct 22 '12 at 4:39
    
It's the other way around: you have more options with large set forcing than with class forcing (all else being equal). However, the underlying consistency assumptions are weaker with class forcing than with large set forcing, but only at face value. –  François G. Dorais Oct 22 '12 at 12:03
    
Do you mean in Con(A_class) -> Con(X) vs Con(A_largeset) -> Con(X), where A_* is the axioms of interest, Con(A_class) is a priori weaker? I'm thinking however of the additional assumptions, like "$\exists$ proper class of x cardinals", that one might make. The alternative is to consider an unbounded set below $\kappa$ of x cardinals, but this would require extra on $\kappa$. But that's another question. –  David Roberts Oct 22 '12 at 19:00
    
The theory NBG is equiconsistent with ZFC but the large set approach assumes that $\kappa$ is inaccessible. At face value, with classes we get that $Con(A) \to Con(B)$ is provable in NBG and with large set we only get that $Con(A) \to Con(B)$ is provable assuming the existence of an inaccessible cardinal. This large cardinal hypothesis is actually not necessary but circumventing it requires some less than natural steps, the least of which is perhaps working in NBG instead. –  François G. Dorais Oct 22 '12 at 19:23
    
Ok, good point. I might ask separately about extra conditions. –  David Roberts Oct 22 '12 at 19:50
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To pin down terms lets say "large set forcing" is what you described above where you assume there is an inaccessible $\kappa$, apply a forcing $P$ of size at least $\kappa$ (i.e. a large set forcing) and then look at $V[G]_{\kappa}$ where $G$ is generic for $P$ over $V$.

Lets then call "class forcing" what is done when you force with a carefully defined class as in, for example, the book "Fine Structure and Class Forcing" by Sy D. Friedman.

There are two issues (as I see them) with large set forcing which class forcing attempts to address.

First, in general it is not the case that with a large set forcing $V[G]_\kappa$ will be a model of ZFC. For example, unless you choose your partial order very carefully, there is no reason to believe that in $V[G]$ that $\kappa$ is even inaccessible. Class forcing gets around this issue by adding conditions on the definable forcing to ensure that when all is said and done the result is still a model of ZFC

The second issue is that in large set forcing you are, by assumption, not dealing with all of the sets. Specifically you can't both assume that there is an inaccessible, use that fact to construct a partial order which wouldn't exist without one, and then say "a ha" there really wasn't an inaccessible.

That being said you are right that very often, especial in category theory, people act as if the universe is really just a set and assume they won't get into trouble. And in fact there are good mathematical reasons for why you can do this (if you are interested I would recommend Feferman's "Set-theoretical foundations of category theory").

Very roughly speaking, there are systems equi-consistent with ZFC which have a class/set distinction where the classes aren't just subclasses of the universe of sets (but rather some not specified combination of subclasses, subsubclasses, etc.) These systems are then able to stay equiconsistent with ZFC because their collection of non-set classes is only required to satisfy a minimal amount of separate/replacement (at least with non-set parameters). A good example of a set theory which does this (in addition to Feferman's mentioned above) is Ackermann's set theory.

What helped me internalize why this collection of non-set classes can't be forced to satisfy many of the separation/replacement axioms which we would like, was the realization that Morse-Kelly Set theory, which differs from Godel-Berney's set theory only in that comprehension is allowed to have set variables, has strictly greater consistency strength.

Anyhow, getting back to class forcing. In some sense one of the main things class forcing is really doing is making sense of how one forces when one does not necessarily have the closure properties which are satisfied by sets (but not by classes).

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What do you mean by $V[G]_\kappa$? I also presume $V$ is the base model rather than the universe. –  David Roberts Oct 22 '12 at 8:16
    
I should say that category theoretically, one just considers a universe $U$ as a small category internal to the ambient set theory, the poset $P$ of forcing conditions as small in the ambient set theory, but large compared to $U$. Then one can consider the category of internal double negation sheaves which is again a small category. This is exactly the sheaf version of forcing, and indeed large set forcing as I have described. But there doesn't appear to be a material set theory (aka ZFC or similar) version of this, or at the very least, an explanation of how this relates to class forcing. –  David Roberts Oct 22 '12 at 8:24
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