# Difference between revisions of "Worldly"

Zetapology (Talk | contribs) m (Added fact of consistency strength) |
Zetapology (Talk | contribs) m (Added replacement characterization) |
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{{DISPLAYTITLE: Worldly cardinal}} | {{DISPLAYTITLE: Worldly cardinal}} | ||

− | A cardinal $\kappa$ is ''worldly'' if $V_\kappa$ is a model of $\text{ | + | 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. | * Every [[inaccessible]] cardinal is worldly. | ||

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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. | ||

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+ | ==Replacement Characterization== | ||

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+ | 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. |

## Latest revision as of 11:02, 30 November 2018

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.