= 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.
Back to article page
