[FOM] Grothendieck fdns, classes in weak set theories

[Colin McLarty]

Thanks for the answers on sequential theories and omega-models.

Now I have a terminological question about "proper classes" in weak
set theories.

With the axiom scheme of replacement every collection of sets the size
of a set is a set.  So it makes sense to say proper classes are the
collections too big to be sets.  And furthermore, the usual proper
classes are things like the collection of all ordinals or all sets or
all groups and so on, which are provably not sets.

But in ZC (Zermelo set theory, including choice) even an explicitly
enumerated collection need not form a set, obviously, with the basic
example of all Aleph-sub-n for finite n.  This is not "too big" but
"too poorly bounded."  And ZC merely fails to prove it is a set,
without proving it is not a set (of course, since standard stronger
set theory proves it is).

So, is it usual to say this collection can be a "proper class" in ZC?
Or is that name reserved for collections which ZC already proves
cannot be sets?  What is the terminology of collections which are not
provably sets in ZC?

And, can anyone recommend a good reference on class models of set
theory?  People working on Reinhardt cardinals in ZF (without choice)
say it makes a real difference what theory of classes you use.   Is
there a good survey of those issues?

thanks, Colin