# Worldly cardinal

A cardinal $\kappa$ is *worldly* if $V_\kappa$ is a model of $\text{ZF}$. It follows that $\kappa$ is a strong limit, a beth fixed point and a fixed point of the enumeration of these, and more.

- Every inaccessible cardinal is worldly.
- Nevertheless, the least worldly cardinal is singular and hence not inaccessible.
- The least worldly cardinal has cofinality $\omega$.
- Indeed, the next worldly cardinal above any ordinal, if any exist, has cofinality $\omega$.
- 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 hyper-inaccessible cardinals. Every 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.