Climb into Cantor’s Attic, where you will find infinities large and small. We aim to provide a comprehensive resource of information about all notions of mathematical infinity.

View the Project on GitHub neugierde/cantors-attic

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The upper attic

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The lower attic

The parlour

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**Sources**

Cantor's Attic (original site)

Joel David Hamkins blog post about the Attic

Latest working snapshot at the wayback machine

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.

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.

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.