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 On into the proper class W, so that we now know all about the six injection possibilities between 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 NBG where the proper class P(On) is such that:

(i) Evidently On injects in P(On)that injects in P(P(on))=V;

(ii) But V=P(P(On)) does not inject in P(On), that itself does not inject in On, so that the Proper Cardinal level P(On)* is distinct from both On* and V*;

(iii) Moreover there can be no injection from P(On) into W, because we could chain with an injection from On into P(On) and build an injection from On into W, that is impossible. So that W*, that is already distinct from the distinct proper Cardinal levels On* and V*, is also distinct from P(On)*.

Then we see that Ali Enayat's model of 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(On) ?

Concerning surjections, we have that:

(i) V surjects onto P(On) that itself surjects onto On;

(ii) V surjects onto W that itself surjects onto On.

Question 3: Is it possible to build a surjection from P(On) onto W, or a surjection of W onto P(on), so that On, P(On), W and V are linearly ordered by surjection ?

Question 4: Is it possible to have more than four distinct Proper Cardinal levels in NBG ?

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 $\mathrm{NBG}$?

This is the continuation of part 1, where all useful definitions and notations are given. J. D. Hamkins answered to the question 1 in this first part, proving that there can be no injection of the proper class On into the proper class W, so that we now know all about the six injection possibilities between On, W and V. Part 2 is about the answer to my question "Bijective-equivalent collections of proper classes in set theory" given by Ali Enayat on14/03/2013. He proved that there exists a model of NBG where the proper class P(On) is such that: (i) Evidently On injects in P(On)that injects in P(P(on))=V; (ii) But V=P(P(On)) does not inject in P(On), that itself does not inject in On, so that the Proper Cardinal level P(On)* is distinct from both On* and V*; (iii) Moreover there can be no injection from P(On) into W, because we could chain with an injection from On into P(On) and build an injection from On into W, that is impossible. So that W*, that is already distinct from the distinct proper Cardinal levels On* and V*, is also distinct from P(On)*. Then we see that Ali Enayat's model of 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(On) ?

Concerning surjections, we have that: (i) V surjects onto P(On) that itself surjects onto On; (ii) V surjects onto W that itself surjects onto On.

Question 3: Is it possible to build a surjection from P(On) onto W, or a surjection of W onto P(on), so that On, P(On), W and V are linearly ordered by surjection ?

Question 4: Is it possible to have more than four distinct Proper Cardinal levels in NBG ?

Gérard Lang

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 On into the proper class W, so that we now know all about the six injection possibilities between 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 NBG where the proper class P(On) is such that:

(i) Evidently On injects in P(On)that injects in P(P(on))=V;

(ii) But V=P(P(On)) does not inject in P(On), that itself does not inject in On, so that the Proper Cardinal level P(On)* is distinct from both On* and V*;

(iii) Moreover there can be no injection from P(On) into W, because we could chain with an injection from On into P(On) and build an injection from On into W, that is impossible. So that W*, that is already distinct from the distinct proper Cardinal levels On* and V*, is also distinct from P(On)*.

Then we see that Ali Enayat's model of 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(On) ?

Concerning surjections, we have that:

(i) V surjects onto P(On) that itself surjects onto On;

(ii) V surjects onto W that itself surjects onto On.

Question 3: Is it possible to build a surjection from P(On) onto W, or a surjection of W onto P(on), so that On, P(On), W and V are linearly ordered by surjection ?

Question 4: Is it possible to have more than four distinct Proper Cardinal levels in NBG ?