We work in the set theory NBG (with local choice, but not global choice), because if there is global choice, every proper class is well-ordered, so that every proper class is bijective with the class On of ordinals and the class V universe, and we only have one level of bijective equivalent proper classes.


As it is known that the Schroeder-Bernstein is valid for proper classes, we have a proper class bijection between two proper classes A and B if we have an injection of A into B and also an injection of B into A, so that we have bijective-equivalent levels of proper classes and that we have a preorder on proper classes corresponding to injections.


From the answer given by Ali Enayat on 15/03/2013 to a question of mine, it is possible to have at least three distinct and ordered levels, namely the level of On, the level of P(On) and the level of P(P(On)), that is also the level of V.


From the answer of JD Hamkins today to another question of me, it is not true that the class On can be injected in every proper class (I hoped that the level of On could be the minimal level in the order on proper classes). This does not completely prove that we have a fourth class under On, because I do not know if it is true that given two proper classes A and B either A injects in to B or B injects into A.


**Question 1:** Is it true that given two proper classes there is an injection between them?


**Question 2:** If the answer to question 1 is true, is there a minimal level for the proper classes levels under injection?



Order in bijective-equivalent collections of proper classes in set-theory
The answer to question 2 is Negative. This is because, answering to my question "Injection of the proper class of ordinals into every proper class", J.D. Hamkins proved on 02/12/2014 the existence in NBG of a proper class W that does not inject into On.He also proved that On does not inject into W.
So that if A were to be the minimal proper class for injection into proper classes, A would inject into On, so would then be bijective with On and well-orderable. But A would also be injective into W; but by chaining an injection from On into A and an injection from A into W, we would get an injection from On into W, that is impossible.
Gérard Lang


