Bijective-equivalent collections of proper classes in set theory In ZFC set theory (or better in NBG set theory, where the language is more flexible with proper classes), we have that every unbounded class of ordinal numbers is a proper subclass of the class On of all ordinals and that every such proper class has a unique bijection (by the enumeration function) with the proper class On. More generally, every proper class that is (class) well-orderable (that is, that is well-orderable, with every descending section being a set) is also bijective with On.
So that, every proper class is bijective with On iff V (the class of all sets) is (class) well-orderable (this is known as the global choice axiom). So, if we do not have the global choice axiom there is no bijection between On and V, and we are left with at least two bijective-equivalent collections of proper classes.
Question 1: Does ZFC (or NBG) prove more that that concerning the bijective-equivalent collections of proper classes  ?
Question 2: If we pass to a set theory whose language allows for quantification over proper classes (like KM, the Kelley-Morse set theory), do we know more about these bijective-equivalent collections of proper classes ?.
Gérard Lang
 A: I will give a partial answer for models of $NBG$ in which local choice ($AC$) holds, namely:

Theorem. There is a model $M$ of $ZFC$ whose natural expansion $(M,\cal{A})$ to a model of $NBG$ (where $\cal{A}$ is the collection of all parametrically definable classes of $M$) has at least 3 bijection-inequivalent classes, namely: $Ord$ (the class of ordinals), $\cal{P}$$(Ord)$ (the class of all subsets of ordinals), and $V$ (of class of sets).
Moreover, it is a theorem of $NBG$ plus $AC$ that there is a bijection between $V$ and $\cal{P}(\cal{P}$$(Ord))$.

Proof. Let $M$ be a model of $ZFC$ which has a definable class of pairs with no no choice function (see, e.g., the answers given by Hamkins and myself to this MO question).
It is clear that there is no injection from $V$ into $Ord$ that is canonically coded in $\cal{A}$.
We next observe that there is no injection from $V$ into $\cal{P}$$(Ord)$ that is coded in $\cal{A}$. This is because there is an obvious definable linear ordering of $\cal{P}$$ (Ord)$  (by viewing each subset of ordinals as a binary sequence), and the existence of such an injection would contradict the fact that there is no definable choice function on a class of pairs.
Finally, to show that there is no injection from $\cal{P}$$(Ord)$ into $Ord$ in our model of $NBG$, we take advantage of the axiom of choice within $M$. More specifically, recall that in the presence of the axiom of choice, every set is definable from a subset of ordinals, the idea being: given any set $x$, let TC($x$) be the transitive closure of {$x$}. Then by AC there is bijection $f$ between TC($x$) and some ordinal $\alpha$. This allows us to copy the $\in$ relation on TC($x$) to a binary relation $r$ on $\alpha$, and then we can use a canonical pairing function on ordinals to code $r$ as some subset $s$ of ordinals. Then $x$ is definable from $s$ via "$x$ is the top element of the transitive collapse of $r$".
Note that the above procedure gives rise to a definable surjection $F$ from $\cal{P}$$(Ord)$ onto $V$: given a subset $s$ of ordinals, first decode it as a binary relation $r$ on ordinals, and then ask whether $r$ is an extensional, well-founded relation with a top element. If no, let $F(s)$ be $0$, and if yes, then let $F(s)$ be the top element of the transitive collapse of $r$.
Now if there exists a definable injection $F$ from $\cal{P}$$(Ord)$ into $Ord$, then in light of the fact that $\cal{P}$$(Ord)$ would then be definably well-ordered, the above map $F$ could be "inverted" by an injection $G$ from $V$ into $\cal{P}$$(Ord)$, where $G(x)$ is the first $s$ such that $F(s)=x$. This, in turn would enable us to definably well-order $V$, contradiction.
Finally, to verify the last claim of the theorem: the coding explained above allows us to define an injection $H$ from $V$ into $\cal{P}(\cal{P}$$(Ord))$, namely $H(x)$ is the collection of all subsets $s$ of $\kappa$, where $\kappa$ is the cardinality of the transitive closure of {$x$}, such that when $s$ is canonically decoded as a relation $r$ on $\kappa$, then $r$ is an extensional well-founded relation with a top element such that $x$ is equal to the top element of the transitive closure of $r$.

Therefore, since the class version of the Schröder-Bernstein theorem holds in models of $NBG$, there is a bijection between $V$ and $\cal{P}(\cal{P}$$(Ord))$ in models of $NBG$ in which the (local) axiom of choice holds.

