This is the continuation of [part 1 of this question](https://mathoverflow.net/questions/124494/bijective-equivalent-collections-of-proper-classes-in-set-theory?rq=1), where all useful definitions and notations are given. J. D. Hamkins answered question 1 in the first part, proving that there can be no injection of the proper class $\mathrm{On}$ into the proper class $W$, so that we now know all about the six injection possibilities between $\mathrm{On}$, $W$ and $V$.


This part (part 2) is about the answer to my question [Bijective-equivalent collections of proper classes in set theory](https://mathoverflow.net/questions/124494/bijective-equivalent-collections-of-proper-classes-in-set-theory?rq=1) given by Ali Enayat. He proved that there exists a model of $\mathrm{NBG}$ where the proper class $P(\mathrm{On})$ is such that:


**(i)** Evidently $\mathrm{On}$ injects into $P(\mathrm{On})$ that injects into $P(P(\mathrm{On}))=V$;


**(ii)** But $V=P(P(\mathrm{On}))$ does not inject into $P(\mathrm{On})$, that itself does not inject into $\mathrm{On}$, so that the Proper Cardinal level $P(\mathrm{On})^\*$ is distinct from both $\mathrm{On}^\*$ and $V^\*$;


**(iii)** Moreover there can be no injection from $P(\mathrm{On})$ into $W$, because we could chain such an injection with an injection from $\mathrm{On}$ into $P(\mathrm{On})$ and build an injection from $\mathrm{On}$ into $W$, which is impossible. Hence $W^\*$, which is already distinct from the distinct proper Cardinal levels $\mathrm{On}^\*$ and $V^\*$, is also distinct from $P(\mathrm{On})^\*$.


Then we see that Ali Enayat's model of $\mathrm{NBG}$ provides a case with (at least) four distinct Proper Cardinal levels.


**Question 2:** *Is it possible to build an injection from $W$ into $P(\mathrm{On})$ ?*


Concerning surjections, we have that:


**(i)** $V$ surjects onto $P(\mathrm{On})$ that itself surjects onto $\mathrm{On}$;


**(ii)** $V$ surjects onto $W$ that itself surjects onto $\mathrm{On}$.


**Question 3:** Is it possible to build a surjection from $P(\mathrm{On})$ onto $W$, or a surjection from $W$ onto $P(\mathrm{On})$, so that $\mathrm{On}$, $P(\mathrm{On})$, $W$ and $V$ are linearly ordered by surjection ?


**Question 4:** Is it possible to have more than four distinct Proper Cardinal levels in $\mathrm{NBG}$?



More on bijective-equivalent classes in NBG set theory (2)
The answer to question 2 is yes. To see this, it suffices to produce from any well-ordering of some $V\_\alpha$ a set of ordinals, such that the well-ordering can be reconstructed from the set of ordinals. Given a well-ordering of $V\_\alpha$, this ordering has some length $\kappa$, and so there is a relation $E$ on $\kappa$ and an isomorphism $\langle V\_\alpha,\in\rangle\cong\langle\kappa,E\rangle$, which is just the bijection determined by the order-isomorphism between $\leq$ on $V\_\alpha$ and the natural order on $\kappa$. The relation $E$ is a set of pairs of ordinals, and we may by the usual pairing function code all the information of $\langle\kappa,E\rangle$ by a set of ordinals $A$. From $A$ we may recover $E$ and therefore (by the Mostowski collapse) the set $V\_\alpha$ and by the bijection to $\kappa$ we recover the order $\leq$. So this provides a (definable) injective map from $W$ to $P(\text{Ord})$, as desired.


It follows that the answer to question 3 is also affirmative, since from any injection from $W$ to $P(\text{Ord})$ we get a surjection in the other direction. 


