Worldly cardinal

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A cardinal $\kappa$ is worldly if $V_\kappa$ is a model of $\text{ZFC}$. 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 $\omega$.
  • 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 worldly cardinal terminology was introduced in lectures of J. D. Hamkins at the CUNY Graduate Center and at NYU.