I've just begun reading Paul Bernays's Axiomatic Set Theory (1958). In it, I noticed something a bit unfamiliar to me and I was wondering if there was anything substantial to it or if it was just an idiosyncratic manner of exposition on which nothing crucially depends.
Usually when I've seen set theories that admit of classes, the sets have been identified with classes satisfying certain criteria. For example, on Gödel's approach sets simply are those classes that are members of classes. The proper classes are those classes which aren't, themselves, members of a class. On this approach, the relation between sets and "set-like classes" is simply identity.
In Bernays's system, however, he talks of classes as represented by sets:
Still the concepts concerning the relation between classes and sets have to be considered. We say that a set represents a class if both have the same elements....[E]very set represents a class, and a class can be represented only by one set. But it does not result that every class is represented by a set. (p. 63)
In the historical introduction to Bernays (1958), Fraenkel notes a key difference between the theories of von Neumann and Gödel and that of Bernays:
[W]ith respect to the Axiom of Subsets [i.e., Separation], an ingenious device enables Bernays to dispense with the general concept of predicate or function in his class axiom...: For every predicate $F$ which contains no bound class-variables, there exists the class of those sets which satisfy $F$. This seems to be an axiom schema, but is (in Bernays' axiom group III) reduced to a small number of single axioms for the construction of classes. Hereby also the Axiom of Substitution [i.e., Replacement] can be transformed to a single axiom. (p. 33)
Bernays claims this as a motivation in the introduction:
As is well known, it was the idea of Frege to identify sets with extensions of predicates and to treat these extensions on the same level as individuals. That this idea cannot be maintained was shown by the Russell paradox.
Now one way to escape the difficulty is to distinguish different kinds of individuals and thus to abandon Frege’s second assumption; that is the method of type theory. But another way is to give up Frege’s first assumption, that is to distinguish classes as extensions from sets as individuals. Then we have the advantage that the operation of forming a class {x | U(x)} from a predicate U(c) can be taken as an unrestricted logical operation, not depending on a specifying comprehension axiom. (p. 42)
Is there anything crucial that hinges on this, though? It seems that he wants class formation to count as "purely logical" and so bifurcates collections into sets and classes. But does this require sets to merely "represent" classes? Is there more to the distinction besides classes being formed through a "purely logical operation"?
For example, does Bernays have an intensional conception of classes such that which set represents a given class can vary according to the interpretation? I'm thinking of a view similar to the one advanced by William Reinhardt (who cites one of Bernays's German papers extensively; but I don't read German....) in his "Remarks on Reflection Principles, Large Cardinals, and Elementary Embeddings" (1974). On this view the crucial distinction between sets and classes is that classes could have different members under counterfactual conditions:
A proper class $P$ may however be distinguished from a set $x$ in the following way...: If there were more ordinals..., $x$ would have exactly the same members, whereas $P$ would necessarily have new elements. We could say that the extension of $x$ is fixed but that of $P$ depends on what sets exist. Roughly, $x$ is its extension, whereas $P$ has more to it than that. (p. 196)
Does Bernays have a similar distinction in mind, which leads to his talk of "representation" of a class by a set? More generally is there anything beyond a desire to keep class formation as a "purely logical operation" that motivates or necessitates having sets merely represent "set-like" classes, instead of identify them?
We can see Gert Muller's Introduction to Sets and Classes: on the Work by Paul Bernays (1976), pag.vii:
The key-point, IMO, is in the paragraph referencing to Frege; the "underlying logic" has predicates $\mathfrak P(x)$ and for every predicate we have its extension $\{ x \mid \mathfrak P(x) \}$.
Extensions are, as for Frege, "logical" objects. And then we have mathematical objects: the sets
Not every logical object is a mathematical one. An extension that is not a set cannot be treated as "one object", i.e. it cannot stand to the left of the $\in$ sign.
See also: Akihiro Kanamori, Bernays and Set Theory, BSL (2009).