About a representation on cardinals

Question

If k is a limit cardinal, necessarly one have that $ ZFC ^ {R_{K}} $ ?

And, if k is only ordinal, does it have any sense?

Answer 1

If $\kappa$ is not a strong limit cardinal, then the answer is clearly not. Because in that case there is some $\alpha<\kappa$ such that $R_\alpha\geq\kappa$, in which case $R_\kappa\models R_\alpha\text{ cannot be put in bijection with an ordinal}$.

If $\kappa$ is a strong limit cardinal, but not a fixed point of the $\beth$ function then $\kappa=\beth_\alpha$ and $\alpha<\kappa$. In that case we have a definable function whose domain is $\alpha$, but its range is unbounded in the ordinals of $R_\kappa$. So replacement fails.

If $\kappa$ is not a limit cardinal (ordinal or otherwise), then $R_\kappa$ thinks that there is a maximal cardinal, so it doesn't satisfy $\sf ZFC$.

More specifically if $\sf ZFC$ can prove $\kappa$ with a certain property exists, then it doesn't prove that $R_\kappa$ necessarily satisfies $\sf ZFC$. This is true for limit ordinals, cardinals, limit cardinals, strong limit cardinals, and so on. If $\sf ZFC$ can prove it exists, it can't prove that $R_\kappa$ is a model of $\sf ZFC$.