Recall the definition of *[cardinal definable](https://mathoverflow.net/questions/412534/is-every-set-being-cardinal-definable-consistent-with-zf)*, where every set being cardinal definable is [proved](https://mathoverflow.net/questions/412534/is-every-set-being-cardinal-definable-consistent-with-zf/412536#412536) consistent relative to ZF + V=HOD. To re-iterate it:


$Define: X \text { is cardinal definable} \iff \\\exists \text { cardinal } \kappa \, \exists \text { cardinals } \lambda\_1,.., \lambda\_n <^\rho \kappa \ \exists \phi : \\ X=\{ y \in V\_{\rho(\kappa)} \mid \phi^{V\_{\rho (\kappa)}} (y,\lambda\_1,..,\lambda\_n)\}$


Where: $\lambda\_i <^\rho \kappa \iff \rho(\lambda\_i) < \rho(\kappa)$, and $\rho$ is the rank function; and "*cardinal*" is defined after Scott's as an equivalence class under *bijection* of sets of the lowest possible rank.



> 
> Now, is the principle stating that *every set is cardinal definable* consistent with $\sf ZF + \neg AC$?
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A related question replacing *cardinal* by *ordinal* in the above question would lead to $\sf V=HOD$, which is known to prove $\sf AC$ and so would be inconsistent with $\sf ZF + \neg AC$.



Is every set being cardinal definable consistent with ZF + negation of Choice?
This is consistent. Kanovei constructed a model $M$ with an infinite Dedekind finite set of reals which is lightface projectively definable. By descending to $L(R),$ we can further assume it satisfies $V=L(R).$


Clearly choice fails in this model. Since it satisfies $V=L(R),$ every set is definable from an ordinal and a real. Of course, any ordinal is definable from an $\aleph,$ so we just need to check that every real is cardinal definable.


Let $A$ be the definable Dedekind finite set of reals, and fix a canonical surjection $f: A \rightarrow \omega$ (as exists for any infinite set of reals). Let $\langle q\_n \rangle$ be an enumeration of the rationals. Define $A\_r=f^{-1}(\{n: q\_n<r\}).$ For $r<s,$ $|A\_r|<|A\_s|$ since these are Dedekind finite sets. Clearly $r$ is definable from the cardinality of $A\_r,$ so we're done.


Reference:


*Kanovej, V. G.*, [**On the nonemptiness of classes in axiomatic set theory**](http://dx.doi.org/10.1070/IM1978v012n03ABEH001997), Math. USSR, Izv. 12, 507-535 (1978). [ZBL0427.03044](https://zbmath.org/?q=an:0427.03044).


