representants of equivalence classes in a proper class Suppose that $\sim$ is an equivalence relation on a proper class $C$. Under what circumstances is it possible to proof the existence of a class $R$ of representatives (i.e., for all $x \in X$ there is a unique $r \in R$ such that $x \sim r$)?
If the equivalence classes are all sets, then the axiom of global choice yields such a class $R$ I think, but in all other cases, I don't know how to proceed.

There are two options:

*

*There is some $\alpha$ such that every $x\in X$ is equivalent to some $r\in V_\alpha$. In that case we normally treat $R$ as somehow having "set many equivalence classes" (e.g. algebraic field extensions or metric completions have these properties). Then you need only the axiom of choice for sets.


*There is no such $\alpha$. In that case, we use the same trick, Scott's trick. Define $[x]_\sim$ to be the elements of $C$ equivalent to $x$ which have the least possible rank. Namely, let $\alpha$ be the least such that for some $r\in V_\alpha$, $x\sim r$, and $[x]_\sim=\{r\in V_\alpha\mid x\sim r\}$.
Now $\{[x]_\sim\mid x\in C\}$ is a class of sets, and the axiom of global choice can be applied.
