A cardinal $\kappa$ is *worldly* if $V_\kappa$ is a model of $\text{ZF}$. It follows that $\kappa$ is a [[Beth#Strong limit cardinal|strong limit]], a <a href="Beth_fixed_point" class="mw-redirect" title="Beth fixed point">beth fixed point</a> and a fixed point of the enumeration of these, and more. - Every [inaccessible](Inaccessible.md "Inaccessible") cardinal is worldly. - Nevertheless, the least worldly cardinal is <a href="Singular" class="mw-redirect" title="Singular">singular</a> and hence not [inaccessible](Inaccessible.md "Inaccessible"). - The least worldly cardinal has <a href="Cofinality" class="mw-redirect" title="Cofinality">cofinality</a> $\omega$. - Indeed, the next worldly cardinal above any ordinal, if any exist, has <a href="Cofinality" class="mw-redirect" title="Cofinality">cofinality</a> $\omega$. - Any worldly cardinal $\kappa$ of uncountable cofinality is a limit of $\kappa$ many worldly cardinals. ## Degrees of worldliness A cardinal $\kappa$ is *$1$-worldly* if it is worldly and a limit of worldly cardinals. More generally, $\kappa$ is *$\alpha$-worldly* if it is worldly and for every $\beta\lt\alpha$, the $\beta$-worldly cardinals are unbounded in $\kappa$. The cardinal $\kappa$ is *hyper-worldly* if it is $\kappa$-worldly. One may proceed to define notions of $\alpha$-hyper-worldly and $\alpha$-hyper${}^\beta$-worldly in analogy with the [](Inaccessible.md#hyper-inaccessible). Every [inaccessible](Inaccessible.md "Inaccessible") cardinal $\kappa$ is hyper${}^\kappa$-worldly, and a limit of such kinds of cardinals. The consistency strength of a $1$-worldly cardinal is stronger than that of a worldly cardinal, the consistency strength of a $2$-worldly cardinal is stronger than that of a $1$-worldly cardinal, etc. The worldly cardinal terminology was introduced in lectures of J. D. Hamkins at the CUNY Graduate Center and at NYU. ## Replacement Characterization As long as $\kappa$ is an uncountable cardinal, $V_\kappa$ already satisfies $\text{ZF}^-$ ($\text{ZF}$ without the axiom schema of replacement). So, $\kappa$ is worldly if and only if $\kappa$ is uncountable and $V_\kappa$ satisfies the axiom schema of replacement. More analytically, $\kappa$ is worldly if and only if $\kappa$ is uncountable and for any function $f:A\rightarrow V_\kappa$ definable from parameters in $V_\kappa$ for some $A\in V_\kappa$, $f"A\in V_\kappa$ also.