Is there anything between proper classes and proper conglomerates, in size? That is, not assuming the generalized continuum hypothesis.

4$\begingroup$ My understanding of the GCH is that it speaks only of sizes of sets, not of biggerthansetsize objects. $\endgroup$– LSpiceAug 14 at 20:18

$\begingroup$ @LSpice Directly, it refers to sets but I don't know if it has implications for conglomerates. $\endgroup$– user489625Aug 14 at 21:14
2 Answers
First of all, in ZFC set theory one cannot prove all proper classes have the same size, and consequently it is not fully sensible to refer to "the size of a proper class," since they can have different sizes. The assertion that all proper classes are equinumerous is not provable in ZFC and is the content of the global choice principle, which is equivalent over ZFC to the assertion that the entire settheoretic universe $V$ is equinumerous with the ordinals, which implies that all proper classes are equinumerous. In firstorder set theory, where one has only definable classes, this is equivalent to the existence of a definable wellordering of the universe.
OK, so let's assume you are asking about conglomerates that have intermediate size between the class of all sets and the conglomerate of all classes.
One can easily manufacture such situations. If $\kappa$ is an inaccessible cardinal, for example, then there is a forcing extension preserving this in which $2^\kappa>\kappa^+$, even if the GCH holds below $\kappa$. If we take $V_\kappa$ as the universe of setsthis is a perfectly good ZermeloGrothendieck universethen every proper class will have size $\kappa$, but the metaclass or conglomerate of all classes will have size $2^\kappa$, which is strictly larger than the conglomerate having one wellordered class of each type $\alpha<\kappa^+$, since this has size $\kappa^+$.
Conclusion: even if the GCH holds for sets, it might not hold between classes and conglomerates, and there can be a conglomerate having size strictly between the class of all sets and the conglomerate of all classes.
Meanwhile, it is also easy to produce models in which there is no such intermediate conglomerate. For example, if $\kappa$ is inaccessible in a model of ZFC+GCH, then the classes of $V_\kappa$ will all have size $\kappa$ and the conglomerates will all have size $\kappa^+$.
One can even arrange that there are abundant violations of GCH at the sets, but still it holds at the top for conglomerates. In general, the GCH for sets has no correlation with the corresponding principle at the level of classes and conglomerates. This might be seen as an analogue of the fact that CH is independent of the instances of GCH at higher cardinals, and in general GCH at one cardinal does not generally determine instances of GCH at other cardinals.
So the answer is that it depends on which particular universe and class conception you are using.

$\begingroup$ What about in Von Neumann's set theory? I think he knew the truth about proper classes. $\endgroup$– user489625Aug 15 at 16:13

$\begingroup$ The argument I give works just the same with NBG or indeed even KelleyMorse set theory. The question between classes and conglomerates is simply not settled by GCH for sets. $\endgroup$ Aug 15 at 16:51
As Joel says, the answer to your inquiry depends on the formalization of 'set', 'class', etc. you choose to work with. Further, if I understand your question correctly on an intuitive level, it seems to imply a slight misunderstanding about the difference between 'sets' and 'classes' (as the words are classically used in set theory) and cardinality considerations.
The generalized continuum hypothesis, as it is typically stated, is a claim about the 'set of functions' construction as it relates to the notion of 'cardinality':
Generalized Continuum Hypothesis. For any infinite set $X$, $$(X,2^X)=\emptyset$$ where $(,)$ is interval notation in the cardinal numbers.
We can trivially consider a version of this for 'classes' or 'conglomerates' (whatever stage of the hierarchy this is supposed to denote), and if your formalization is 'hierarchical' (so all sets are classes, all classes are conglomerates, etc.) these will all be stronger claims. We can also ask versions that hold only for 'proper blorps', so they hold 'at the top' but not necessarily down in the sets/classes etc. as illustrated by some of the models in Joel's answer. In formalizations where sets aren't classes (and so on), these claims need not be related.
Generally speaking, we 'run into classes' when we try to consider sets as extensions of predicates that hold for 'too many objects of discourse' to be collected up, for example $$X\notin X.$$ Classes are 'the answer' to this dilemma in the sense that we can consider a class that extends this predicate for all sets, but if we then try to consider a class that extends this predicate for all classes we return to paradox. We can 'resolve' this issue by introducing another stage of collection called 'conglomerates' and allowing conglomerates to extend predicates for classes, but the issue obviously reappears at this stage and requires yet higher notions of 'collection' to resolve.
I get the sense that your question could be formalized more precisely as follows:
Let $T$ be a set theory that admits three stages of 'collection', which we will call 'sets', 'classes', and 'conglomerates' respectively. Call a class (conglomerate) proper iff it is not also a set (class). Define $$\alpha^\uparrow=\sup\{X:X\ \text{is a set}\},$$ $$\alpha^\downarrow=\inf\{X:X\ \text{is a proper class}\},$$ $$\beta^\uparrow=\sup\{X:X\ \text{is a class}\},$$ $$\beta^\downarrow=\inf\{X:X\ \text{is a proper conglomerate}\}.$$ Then we can formalize the
Collection Hypothesis. $$(\alpha^\uparrow,\alpha^\downarrow)=\emptyset,$$ $$(\beta^\uparrow,\beta^\downarrow)=\emptyset.$$
This hypothesis is independent of standard axiomatic systems for set theory, and unrelated to the generalized continuum hypothesis in the sense that assuming $GCH$ or $\neg GCH$ doesn't impact the answer to the collection hypothesis.