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This is the continuation of part 1 of this question, 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 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 $\mathrm{NBG}$?

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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.

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    $\begingroup$ Dear J. D. Hamkins, let me thank you very much. So we now know everything on the possible injections between the four distinct proper Cardinal levels V*, P(On)*,W* and On*, that linearly ordered by surjection. $\endgroup$ – Gérard Lang Sep 7 '17 at 7:29
  • $\begingroup$ It was my pleasure. For your question 4, I seem to recall some results that one can have quite a lot of class cardinalities, but I don't remember how to achieve this. $\endgroup$ – Joel David Hamkins Sep 7 '17 at 14:38
  • $\begingroup$ Let me ad that I am finding your lastt remark is rather thrilling. $\endgroup$ – Gérard Lang Sep 10 '17 at 13:45
  • $\begingroup$ Well, I can't quite recall it, and I am afraid that I may simply be mis-remembering Ali Enayat's very fine results. $\endgroup$ – Joel David Hamkins Sep 10 '17 at 13:48

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