The collection of all groups is a proper class, since every set gives rise to a group. But what about the collection of all isomorphism classes of groups? By which argument do I see, that it is a set or a proper class?



Is the collection of isomorphism classes of groups a proper class?
Since free groups are isomorphic if and only if they have a basis of the same cardinality (probably assuming some choice axiom here), the collection of all isomorphism classes of groups has at least the size of the collection of all isomorphism classes of sets, hence is not a set.


