Difference between revisions of "Worldly"

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{{DISPLAYTITLE: Worldly cardinal}}
 
{{DISPLAYTITLE: Worldly cardinal}}
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.
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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 10:02, 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.