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


Where: $\lambda\_i <^\rho \kappa \iff \rho(\lambda\_i) < \rho(\kappa)$, and $\rho$ is the rank function; and "*cardinal*" is defined after Scott's.



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> Now, is the principle that every set is cardinal definable consistent with ZF?
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Is every set being cardinal definable consistent with ZF?
Yes, assuming ZF is consistent it is, because it follows from ZFC + V=HOD. This is because the ordertype of the class of (ordinal-)cardinals is just that of the ordinals. That is, if $X$ is definable over $V\_\alpha$ from ordinal parameter $\beta<\alpha$, then $X$ is definable over $V\_{\aleph\_\alpha}$ from ordinal parameter $\aleph\_\beta$, since $\xi\mapsto\aleph\_\xi$ restricted to $\alpha$ is also definable over $V\_{\aleph\_\alpha}$.


