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 $\mathrm{NBG}$?


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.

  • 1
    $\begingroup$ 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

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.