In an old discussion thread at sci.math, Herman Rubin said:
"
There exist models where all proper classes have the same cardinality; i.e., the universe is equinumerous with the class of ordinal numbers.
There exist models where there are proper classes which are neither larger nor smaller than the class of all ordinal numbers.
I am almost certain that there exist models with proper classes strictly larger than the class of all ordinal numbers, and all comparable.
"
On the other hand Asaf Karagila in a comment to this posting to MO, have said:
"
in ZF we can at least prove that every subset of $V_\kappa$ is an element of it, or maps onto $\kappa$.
" ($\kappa$ is meant to be inaccessible).
I take that to mean in absence of choice [since its stated in ZF alone]. Now in some sense $V_\kappa$ can be viewed as being the standard model of MK.
Clearly the second and third of Rubin's statements indicate the existence of a model of a variant of MK\NBG in which there do exist a proper class that is strictly less in size than $V$ namely the class $\small \sf [ON]$ of all ordinal numbers (that are sets of course).
I'm interested in the last of Rubin's statements. He said "all being comparable". I don't know if that entails choice [not global of course].
In the same referred thread Aatu Koskensilta answeres to the third of Rubin's statements by:
"
Using the consistency of ZFC + "there is an inaccessible" (and standard well known independence results) it's easy to show there are such models. As noted, by some tweaking we can remove the inessential inaccessible, bringing us back to ZFC and MK in terms of consistency strength.
"
Questions:
What are the models of those variants of MK in which there are proper classes of sizes strictly smaller than the universe $V$ of all sets, that are closest to the standard model of MK. Provided that the total number of proper classes in them is higher than that of sets.
Is there a restrain on the total number of such proper classes (i.e. strictly smaller than $V$) in such models.
Can those models satisfy choice for sets.
