Locally small category whose collection of isomorphism classes cannot be a set

Question

For natural examples of locally small categories like the category $\mathbf{Grp}$ of groups, the isomorphism classes themselves are normally not sets. In a set theory like ZFC, even the collection of a finite number of such isomorphism classes would not be a set, because the members of a set must be sets themselves.

But suppose we had a set theory with ur-elements and classes, which would allow us to form such collections, as long as they are well enough defined. For some categories like the category $\mathbf{FinSet}$ of finite sets, the collection of isomorphism classes will form a nice countable set. I'm not sure what happens for the category of (hereditarily) countable sets. (Maybe the corresponding collection will still be countable?)

For a category like $\mathbf{Grp}$, the collection of isomorphism classes will probably not be countable, but instead will probably be quite "huge" relative to the available set universe. I'm not sure whether it will "always" be a set, but still this collection seems to be much smaller to me than any single isomorphism class of this category. But how can we distinguish a natural category like $\mathbf{Grp}$ where the collection of isomorphism classes is still "reasonably small" from unnatural categories where the notion of isomorphism is too restricted, such that the collection of isomorphism classes cannot be a set, even for the most powerful set theories?

Answer 1

One way to circumvent this kind of issue is to work with Grothendieck's universes introduced in SGA4 (as an appendix by Bourbaki if I remember correctly).

For any universe $\mathbb U$, define $\mathbb U$-categories as categories where the hom-sets are elements of $\mathbb U$. So you have the categories $\mathbb U{-}\mathsf{Set}$, $\mathbb U{-}\mathsf{Grp}$, etc.

Then the most important axiom of Grothendieck universes is : for any naive set, there exists a universe containing it. Apply it to, say, the collection $\operatorname{Ob}(\mathbb U{-}\mathsf{Grp})$ which is in some universe $\mathbb V$ : the category $\mathbb U{-}\mathsf{Grp}$ is then a small $\mathbb V$-category.

In that framework, there is no unnatural category !

Answer 2

While ZF and ZFC are not really expressive enough for even properly stating my question, the conservative extension NBG works pretty well for this question, at least as far as classes are concerned.

• The isomorphism classes are normal classes in the sense of NBG.
• In a set theory with ur-elements, it would be nice if it were possible to turn these classes into ur-elements, by forming the corresponding 1-tuples. There are no ur-elements in NBG, but we can select a representant from each isomorphism class instead. This is possible, because we have global choice in NBG.
• The resulting collection is once again a normal class from NBG, so that we can talk about its size. Because we have limitation of size in NBG, all classes which are not sets have the same size. As pointed out by Zhen Lin, the collection of isomorphism classes of $\mathbf{Grp}$ has the same size as the class of all cardinal numbers. A single isomorphism class can't be bigger than that, which seems to answer my question in the negative sense.
• Using TG instead of NBG doesn't really change the state of affairs related to this question. Hence even so I learned a bit from Pece's answer, it is ultimatively misleading. This is because the consistency strength of ZF is not really the issue. It's a different sort of expressive power, which is needed here. Being able to talk about classes is one part. Having ur-elements might be the other part (for my intuition), but that would still be a material set theory. This might also be an advantage in a certain sense, but another approach might be to look at a structural set theory. It's sometimes nice to read something new with some concrete questions in mind.

I'm not sure that anybody has really understood my real question. All the answers just concern the set theoretic difficulties, but the question is also tagged as "mathematical-modeling". Yes, I wonder how one can distinguish "correct" modeling of isomorphisms from "evil" equality based modeling. The answers relative to the discussed set theories seem to indicate that it is not possible to distinguish it. However, I'm still not convinced that this is not just an artifact of the reduced ontology of these set theories.