# Worldly

From Cantor's Attic

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A cardinal $\kappa$ is *worldly* if $V_\kappa$ is a model of ZFC. 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 $\omega$.

## 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.