how many empty sets are there? Would I be correct in saying that in the category of sets, the "class of sets that are isomorphic to the empty set is a proper class"?
In other words, there are LOTS of initial objects in the category of sets, but they're all related to each other via unique isomorphisms?
Similarly, for any category $C$, if $X$ is an object of $C$, then the full subcategory of objects isomorphic to $X$ is a proper class (ie, there are LOTS of them)?
EDIT: Let me elaborate a bit. I'm not entirely sure if I should be considering ZFC or ETCS, etc. I'm asking this question from the point of view of algebraic geometry and its use of category theory. I mean, I know people say that the class of say, all elliptic curves over $\mathbb{C}$ is a proper class, but the class of $\mathbb{C}$-isomorphism classes of elliptic curves is a set. That would seem to imply that every isomorphism class is a proper class, and hence this would seem to say that the isomorphism class (ie, bijection class) of the empty set in the category of sets should also be a proper class, right?
Lana's answer with the extensionality axiom seems to make sense, in that there is only ONE empty set, so that the isomorphism class of the empty set is a singleton set. On the other hand, the isomorphism class of a set with 1 element should be a proper class right?
Generalizing a bit, would it be true that the isomorphism class of any initial (or terminal) object of a category be a singleton set? It seems to me that the isomorphism class of, say $\text{Spec }\mathbb{Z}$ in the category of schemes is a proper class, simply because there are many different presentations of rings isomorphic to $\mathbb{Z}$?


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 A: By extensionality axiom you get that there exists exactly one empty set, and that's all folks!
A: Your conclusion is not entirely correct. While it's perfectly fine to have a context where a category has many initial objects which are isomorphic (but not equal), there is a delicate point to the inaccurate statement:

the class of say, all elliptic curves over $\mathbb{C}$ is a proper class, but the class of $\mathbb{C}$-isomorphism classes of elliptic curves is a set. That would seem to imply that every isomorphism class is a proper class

The fact that there is a proper class of different structure is quite trivial to show, because we can always change one element with another and get a proper class of different, but isomorphic, structures that way.
To say that there is only set-many isomorphism classes is also inaccurate, because the isomorphism classes are proper classes, there is no collection of the form $\{A\mid A\text{ is an isomorphism class of ...}\}$. Only sets can be members of other sets. But it does mean something else, it means that there is a set of pairwise non-isomorphic structures that any other structure is isomorphic to one of them. That is, there is a class of representatives which is a set.
Lastly, it does not mean that every equivalence class is a proper class, just that at least one of the equivalence classes are.
The simplest example of this is equinumerousity (in $\sf ZFC$). There is a proper class of sets of every cardinality, except for the class of sets of cardinality zero. There is only one of those.
If your foundational theory is $\sf ZFC$ or some related theory then there exists only one empty set. If your foundational theory is some category based theory which allows many empty sets, but requires them to be isomorphic, then this is a different case altogether.
A: Let $A$ and $B$ be empty sets, then by logic (namely when if-statements are true) $A \subset B$ and $B\subset A$, so $A = B$ (since $A\subset B$ iff $x\in A\implies x\in B$, but x is not in A, so it is vacuously true, same for $B\subset A$). So there is only one initial object in $\textbf{Set}$ up to equality, not just isomorphism.
Edit: "On the other hand, the isomorphism class of a set with 1 element should be a proper class right?" Yes because the class of sets is a proper class and you can take all of the sets of the form $\{A\}$ where A is a set.
"Generalizing a bit, would it be true that the isomorphism class of any initial (or terminal) object of a category be a singleton set?" No because the objects in a general category need not be just sets.
