Can one combine proper classes into a set?

Question

Let's assume you have two non-set classes $C_1$ and $C_2$ of objects. (E.g., $C_i$ is the class of all algebras of some distinct signature $S_i$ for $i=1,2$.) Now, is there a reasonable variant of class logics and set theory where the class $\{C_1,C_2\}$ is (perhaps, under certain restrictions) a set containing exactly two members?

Answer 1

Matters largely depend on how you define set class and non-set class. For example lets take ZFC+ there exists exactly one inaccessible cardinal $\kappa$. Now in this set theory name every object in it as class, and name any class that is equi-numerouse to an element of $V_{\kappa}$ as set, and any class that is bigger than all elements of $V_{\kappa}$ to be a non-set class. Then you'll have a theory with your requirement.

Moreover, I think you can do that in Ackermann's set theory, without being involved with the assumption of existence of inaccessibles, so you remain within the strength of $ZFC$ over the realm of sets of this theory. You can easily upgrade the class comprehension axiom to a class separation axiom after Muller's, apply the same re-definitions given above [with $V$ instead of $V_{\kappa}$], and you'll get similar results.

This definition sets the size of a class as a definition cut-off criterion between what is a set and what is not.

Even a wilder approach is to define a class as non-set if and only if it is of inaccessible cardinality. So according to this definition you can even have sets having subclasses of them that are non-sets.