On 3/29/2013 5:24 PM, Shmuel (Seymour J.) Metz wrote: > In <firstname.lastname@example.org>, on 03/29/2013 > at 06:07 AM, quasi <email@example.com> said: > >> There are certain concepts for which sets are inadequate and classes >> come to the rescue. For example, we need the class concept if we >> want to define an equivalence on the collection of all groups, since >> that collection is not a set. > > There's no such set in ZFC, but there are set theories in which it > exists aqnd is a set. >
Can you name one? I am curious.
I had begun dusting off my category theory books to address that aspect of the question. MacLane defines metacategories and then instantiates categories relative to set representations. The result is a distinction between large categories and small categories.
Then there is the use of Grothendieck universes. Those are essentially inaccessible cardinals as described by a particular form introduced by Tarski. But, even with these, there is an implied hierarchy of universes for which no mathematics has been known to need more than a sequence of 3 (per the account that I read).
I suppose in that regard, there may be
(some standard set theory)+(some large cardinal axiom)
combination that constitutes a set theory different from ZFC for which your statement holds. Is this what you had in mind?