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Is it consistent with Morse-Kelley set theory without global choice (but with choice for sets) that there are $2^{|V|}$ proper classes of different cardinalities?

Alternative question: Is it consistent with ZF that there is an inaccessible $κ$ such that $V_κ⊨\text{ZFC}$ and there are at least $|V_{κ+1}|$ cardinalities of subsets of $V_κ$?

With global choice, all proper classes are equinumerous with $V$, so the question is whether without global choice the extreme opposite can hold.

Morse-Kelley set theory (MK) has full comprehension for proper classes. One formalization of $2^{|V|}$ cardinalities of classes is existence of a formula (allowing parameters) encoding a function $F$ (with $\operatorname{ran}(F)⊆\operatorname{dom}(F)=P(V)$) such that $∀A \, ∀B \, (A≠B ⇒|F(A)|≠|F(B)|)$. Inconsistency would then be a schema over formulas. However, this formalization has a defect of (in a sense) requiring definability, hence the inclusion of the alternative question above.

I suspect that the answer is yes, even if having different cardinalities is strengthened to incomparability under surjections. Also, for first-order-definable (with set parameters) proper classes, I suspect that having $|V|$ different cardinalities is consistent (for a partial result, see here).

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  • $\begingroup$ The alternative question, yes, easily arranged. Okay, maybe not that easily. $\endgroup$
    – Asaf Karagila
    Commented Oct 9, 2022 at 17:40
  • $\begingroup$ @AsafKaragila As I suspected, despite the initial counterintuitiveness of having so many different cardinalities. When you have time, feel free to post an answer, or if it does not quite work, some comments with partial results. $\endgroup$ Commented Oct 9, 2022 at 18:14

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