 Worldly cardinals below an inaccessible cardinal

Below any uncountable strongly inaccessible cardinal $\kappa$, the worldly cardinals contain a closed unbounded subset of $\kappa$ and hence have cardinality $\kappa$. The structure $V_{\kappa }$ models ZFC. Reflect each finite group from an enumeration of all first-order formulas to a closed unbounded subset of $\kappa$, using $|V_{\alpha }|<\kappa$ and regularity for the witness bounds. The countable intersection remains closed unbounded, so its ranks are elementary substructures of $V_{\kappa }$. Intersect with the closed unbounded subset of infinite cardinal numbers below $\kappa$. Uncountability is essential: the regular strong limit $\omega$ has no worldly cardinal below it.