Difference between revisions of "Worldly"
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As long as $\kappa$ is an uncountable cardinal, $V_\kappa$ already satisfies $\text{ZF}\setminus\textrm{Replacement}$ ($\text{ZF}$ 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 $A\in V_\kappa$, $f^{\prime\prime}A\in V_\kappa$ also. | As long as $\kappa$ is an uncountable cardinal, $V_\kappa$ already satisfies $\text{ZF}\setminus\textrm{Replacement}$ ($\text{ZF}$ 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 $A\in V_\kappa$, $f^{\prime\prime}A\in V_\kappa$ also. | ||
| − | == | + | ==Otherworldly cardinals== |
| − | [ | + | J. D. Hamkins has named a large cardinal property called the ''otherworldly cardinals'': $\kappa$ is otherworldly if there exists some $\lambda>\kappa$ such that $V_\kappa\prec V_\lambda$. [http://jdh.hamkins.org/otherwordly-cardinals/ "Otherworldly cardinals" (2020)] |
| + | |||
| + | Otherworldly $\kappa$ have some properties: | ||
| + | * Every otherworldly cardinal is worldly (which played a part in inspiring the name), and also happens to be a limit of worldly cardinals. | ||
| + | * Every otherworldly $\kappa$ is a limit of cardinals $\lambda<\kappa$ such th | ||
Revision as of 12:48, 1 July 2022
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. (See Grothendieck universe)
- 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 $\text{ZF}\setminus\textrm{Replacement}$ ($\text{ZF}$ 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 $A\in V_\kappa$, $f^{\prime\prime}A\in V_\kappa$ also.
==Otherworldly cardinals==
J. D. Hamkins has named a large cardinal property called the otherworldly cardinals: $\kappa$ is otherworldly if there exists some $\lambda>\kappa$ such that $V_\kappa\prec V_\lambda$. "Otherworldly cardinals" (2020)
Otherworldly $\kappa$ have some properties:
- Every otherworldly cardinal is worldly (which played a part in inspiring the name), and also happens to be a limit of worldly cardinals.
- Every otherworldly $\kappa$ is a limit of cardinals $\lambda<\kappa$ such th
