Can we avoid ever needing to mention proper classes in math by restricting to subsets of a universal working set?
More precisely, can we say that anything provable/true with classes is provable/true by taking an arbitrary set U and restricting to subsets of U? Moving towards more of a formalization: let S be a sentence on sets, then is it the case that S is true if and only if for any set U, the corresponding sentence formed by replacing variables ranging over arbitrary sets in S with variables ranging over subsets of U is true?
Let’s have some motivating examples to guide us. Instead of say considering the class of all groups (which is a proper class), we can consider the set of all subsets of U that are groups (the combination of set plus operations on it.) If we imagine a sentence “C is the class of all sets that are groups,” then we can transform that sentence to “C is the set of all subsets of U that are groups.” If we transform all the axioms and such in this way, then do the theorems transform equivalently? In other words, if we have a language involving the symbol C and we replace an axiom “C is the class of all sets that are groups” with “C is the set of all subsets of U that are groups,” then do any theorems about C transform equivalently as well?
I was originally confused here and thought this would have implications for the validity of some of model theory (leading to this question I asked on StackExchange: https://math.stackexchange.com/questions/4818492/model-theory-of-axiomatic-set-class-theories.) I specifically asked: how is it possible for the “collection of all classes” to be a model of NBG if no such collection can exist, since proper classes can’t belong to anything? Well, model theory of NBG just doesn’t apply to all classes; it applies to other models of the axioms (sets or classes that satisfy the NBG axioms), which may have nothing to do with our “traditional” model of sets with their natural membership relation and such. (Natural numbers aren’t the only model of a + b = b + a, even if for a beginner that’s what first comes to mind as a model.) (Also, another note: in model theory, people tend to assume that any model must be set-sized, and they tend to use the term “inner model” or “class model” for a proper class-sized model. See the comments on my question.) So to study the theory of all classes, we just can’t do model theory of NBG. We can certainly try other methods, like proof theory of NBG or whatnot (which again would generally apply to all models of NBG, not just “all classes.”)
I’ve also heard that Grothendieck universes are a preferred way to specifically avoid using proper classes in math; how do they relate to this discussion?
Well, basically, starting with subsets of U and doing whatever operations we want with them (like power set) produces elements of the Grothendieck universe containing U. (We can say “the” by automatically defining this to mean the “smallest” — the relevant considerations should check out without issue.) If we assume the “axiom of Grothendieck universes” that any nonempty set belongs in a Grothendieck universe, we can just assume any set we deal with is a member of some fixed Grothendieck universe, and since we can do this for any set, that would give us a sufficient means to avoid proper classes and have the same results. This is a standard thing already done in the literature. But ZFC can’t prove this axiom, so if we want to only rely on ZFC, we can try to replace Grothendieck universes with subsets of a universal set. (As far as I am currently thinking, this is the only reason to use subsets of a universal set instead of a Grothendieck universe.) However, intuitively/informally speaking, anyway the axiom of Grothendieck universes is a “reasonable” axiom, just as reasonable as any of the axioms of ZFC, so we can just assume it and move on.
The main difference between Grothendieck universes and universal sets is that we can end up producing sets that are not subsets of the universal set, using standard set-theoretic operations like the power set. If we want our statements about all sets to apply to all these sets too, not just subsets of the universal set, then that entails working in a Grothendieck universe. So for maximum “power” and generality of our statements, we can just do the standard thing and work in a Grothendieck universe if we want to avoid proper classes.
