A cardinal $\kappa$ is said to be worldly if $V_\kappa$ is a model of ZFC. Let us (potentially) generalize this by saying an ordinal $\alpha$ is worldly if $V_\alpha$ is a model of ZFC. The existence of a worldly ordinal implies the existence of a transitive model of ZFC, hence a countable ordinal $\alpha$ such that $L_\alpha$ is a model of ZFC. But I'm not sure in that case if there must then be a worldly cardinal.
My questions are as follows.
- Does ZFC prove that, if there is a worldly ordinal, then there is a worldly cardinal?
- Does ZFC prove that every worldly ordinal is a (worldly) cardinal?
- If not, is it consistent with ZFC that every worldly ordinal is a cardinal?
Hopefully it's my last set theory question for a while. :)