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A strongly inaccessible cardinal is an uncountable regular cardinal $\kappa$ that is also a strong limit cardinal:

If transitive models of enough set theory contain the same $V_{\alpha +1}$, then they agree on whether $\alpha$ is a strongly inaccessible cardinal. They have the same subsets and functions on every ordinal below $\alpha$, so they agree on cardinality, regularity, and the strong limit cardinal property.

A weakly inaccessible cardinal is an uncountable regular cardinal that is a limit cardinal. It need not be a strong limit cardinal. Under the Generalized continuum hypothesis, it is a strongly inaccessible cardinal, since every smaller infinite $\lambda$ satisfies $2^{\lambda }={\lambda }^{+}<\kappa$.

A cardinal $\kappa$ is worldly when $V_{\kappa }\models \mathrm{ZFC}$. Every strongly inaccessible cardinal is worldly, but worldly cardinals can be singular.

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.