# Difference between revisions of "Worldly"

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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#hyper-inaccessible | hyper-inaccessible cardinals]]. Every [[inaccessible]] cardinal $\kappa$ is hyper${}^\kappa$-worldly, and a limit of such kinds of cardinals. | 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#hyper-inaccessible | hyper-inaccessible cardinals]]. Every [[inaccessible]] cardinal $\kappa$ is hyper${}^\kappa$-worldly, and a limit of such kinds of cardinals. | ||

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+ | 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. | The worldly cardinal terminology was introduced in lectures of J. D. Hamkins at the CUNY Graduate Center and at NYU. |

## Revision as of 13:28, 14 December 2017

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 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.