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We work in the NBG set theory (with local choice, but without global choice).

$V=\{x∣x=x\}$ is the universal class and $\emptyset=\{x∣ not x=x\}$ is the empty class. Every class $A$ satisfies $A\subseteq V$; $A$ is a proper class if $A\notin V$ and a set if $A\in V$. For every class the power set of $A$ is $P(A)=\{x∣x\subseteq A\}$. We write $AbB$ if we can prove the existence of a (maybe class) function that is bijective between $A$ and $B$. By replacement there can be no bijection between a set and a proper class. We name Cardinal level an equivalence class of classes for the relation $i$ on the collection of all classes, where Cardinal level is an i-equivalence level with only sets and Proper Cardinal level is an $i$-equivalence level with only proper classes. $A^*$ is the the $i$-equivalence level of the class $A$.

We write $A^*JB^*$ if every class in $A^*$ injects into every class in $B^*$. For every set $x$ and every proper class $A$ we have $x^*JA^*$. The relation $J$ is reflexive and transitive. Moreover, the Cantor-Bernstein theorem extended to classes proves that $J$ is antisymmetric and thus an order relation, where $V^*$ is maximal and $\emptyset^*$ is minimal, and whose restriction to cardinal levels is a linear order.

We write $A^*SB^*$ if every class in $A^*$ surjects onto every class in $B^*$. The relation $S$ is reflexive and transitive and thus a preorder. Moreover, from every injection from $A$ into $B$ we can build a surjection from $B$ onto $A$ (we remark that the empty function is an injection from $\emptyset$ into every class and a surjection from every class onto $\emptyset$), so that $A^*JB^*$ implies $B^*SA^*$ and $S$ is finer that the converse relation of $J.V^*$ is minimal and $\emptyset^*$ is maximal in $S$, whose restriction to cardinal levels is the opposite linear order of the restriction of $J$.

With the global choice axiom every pair of classes is bijective (specifically $\mathrm{On}$ and $V$), so that we only have one unique Proper Cardinal level. In fact if every pair of classes is bijective, the axiom of global choice holds. So without global choice we must have at least two Proper Cardinal levels, specifically $\mathrm{On}^*$ (that is the level of all well-orderable proper classes) and $V^*$ (that is the level of all proper classes verifying $P(A)=A$), so that $V^*J \mathrm{On}^*$ is false.

For every ordinal number $a$, using the axiom of foundation, let now $V(a)$ be the set of all sets whose rank is $\leq a$, $W(a)$ be the set of all well-orders on $V(a)$ and $W$ be the proper class that is the union of the sets $W(a)$ indexed by $\mathrm{On}$. J. D. Hamkins proved that there is no injection from $W$ into $\mathrm{On}$ and no injection of $V$ into $W$, so that $W^*$ is necessarily a third distinct Proper Class level and that $J$ cannot be a linear order relation.

But the function $F$ from $W$ onto $\mathrm{On}$ defined by $w\in W(a)\Rightarrow F(w)=a$ is clearly a surjection, so that the restriction of $S$ to $V^*$, $W^*$ and $\mathrm{On}^*$ is linear order.

Question: Is it possible to build an injection from $\mathrm{On}$ into $W$ without global choice?

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  • $\begingroup$ Wow, how about some paragraph breaks? $\endgroup$ – Asaf Karagila Sep 6 '17 at 15:58
  • $\begingroup$ Dear Asaf Karaglia, I completely agree with your remark. But I thought that it was necessary to carefully specify all concepts and notations involved and I feared not to be able to complete my first message if there was limit of place for such messages. So I managed to be as compact as possible and hoped to remain readable. But I certainly will be better in my second message ! $\endgroup$ – Gérard Lang Sep 6 '17 at 17:44
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The answer is no, because the existence of an injection of Ord into $W$ implies global choice. From your definitions, $W$ is the proper class of all well-orders of the rank-initial segments of the universe. Your $V(\alpha)$ is what is usually denoted $V_{\alpha+1}$. Suppose that we had an injection of Ord into $W$. Since for each ordinal $\alpha$, there are only a set of well-orders of $V_{\alpha+1}$, it follows that the range of the injection must eventually include well-orders of arbitrarily large $V_{\alpha+1}$'s. Thus, we may use the injection to define a global well-order as follows: let $x\leq y$ if the rank of $x$ is less than the rank of $y$, or else they have the same rank $\alpha$ and for the least ordinal $\beta$ mapping to a well-order $\leq_\beta$ in which both $x$ and $y$ appear, then $x\leq_\beta y$. This is a well-order of the universe, since any set will have a least-rank element, and then a $\leq_\beta$-least element in that least rank, where $\beta$ is the least ordinal being mapped to a well-order large enough to order sets of that rank.

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  • $\begingroup$ Dear J D Hamkins, thank you very much for this clear answer, so that we now have all injections possibilities with On and V concerning your proper class W. I tried to have more space in my two first tries to copy with Asaf's wishes, but I completely failed ! $\endgroup$ – Gérard Lang Sep 6 '17 at 17:49

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