Question about sets and classes I know every set is a class and not the other way around, but can one consider the set of, say, two classes? Is this "well-defined"?
 A: If von Neumann-Bernays-Gödel Set Theory (NBG Set Theory) is presented as a one-sorted theory, then a set is defined as a class that is an element of another class (possibly itself, but this is ruled out by the Axiom of Regularity for NBG Set Theory). A class that is not a set (i.e., one that is not an element of another class) is called a proper class.
Let $ A $ and $ B $ be classes. If there exists a set $ C $ whose elements are precisely $ A $ and $ B $, then by definition, $ A $ and $ B $ are sets. Conversely, by the Axiom of Pairing for NBG Set Theory, if $ A $ and $ B $ are sets, then there exists a set $ C $ whose elements are precisely $ A $ and $ B $. Therefore, a set consisting of precisely two classes is ‘well-defined’ if and only if the two classes are sets.
In order to deal with the case when the two classes are proper, we must go beyond sets and classes by creating a third type of object, called a conglomerate. We require that conglomerates at least satisfy Extensionality, Pairing, Union, Power and the Axiom Schema of Specification. By default, all classes and sets are conglomerates. However, there exist conglomerates that are neither a set nor a class, such as the object $ \{ V \} $, where $ V $ is the class of all sets. We already know that $ V $ is a proper class, so in NBG Set Theory, the object $ \{ V \} $ does not exist. In the theory of conglomerates, however, $ \{ V \} $ is a well-defined conglomerate.
We can therefore view conglomerates as generally being one level higher than classes and two levels higher than sets. This is reminiscent of Russell's theory of types.
By Pairing, one can thus form a conglomerate of two proper classes, which is one way of resolving the problem of ‘pairing’ two proper classes. You may wish to refer to David Murfet's Foundations for Category Theory for a discussion of conglomerates.
One way of avoiding conglomerates is to use inaccessible cardinals. I believe that Murfet talks about this in the same essay, albeit briefly. I have also provided below two other references where conglomerates are mentioned.
References


*

*Osborne, M. Scott. Basic Homological Algebra (Graduate Texts in Mathematics), Springer, 2000.

*Xu, Kongshi. Advances in Chinese Computer Science, Volume 3, World Scientific, 1991, pp. 155-170.
