Worldly cardinal

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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. (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 (to $\lambda$) if there exists some $\lambda>\kappa$ such that $V_\kappa\prec V_\lambda$. "Otherworldly cardinals" (2020). If $\kappa$ is inaccessible and otherworldly to a $\lambda$ that is greater than some ordinal $\theta$, then $\kappa$ is $\theta$-pseudo-uplifting.

$\kappa$ is called otherworldly up to $\lambda$ if the set of $\mu$ such that $\kappa$ is otherworldly to $\mu$ is cofinal in $\lambda$.

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 that $Th(V_\lambda)=Th(V_\kappa)$.
  • Every inaccessible $\delta$ is a limit of otherworldly cardinals.
    • In fact, inaccessible $\delta$ is the supremum of the class $\{\kappa\in\delta\mid V_\kappa\prec V_\delta\}$.
      • $\delta$ is a limit of cardinals otherworldly up to and to $\delta$.
  • A cardinal is otherworldly iff if it is fully correct in a worldly cardinal.

A cardinal $\kappa$ is totally otherworldly if for all $\lambda>\kappa$ we have $V_\kappa\prec V_\lambda$ ($\kappa$ is otherworldly to arbitrarily large ordinals).

  • Every totally otherworldly cardinal is $\Sigma_3$-correct. [1]