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.

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*Hypercompactness* is a large cardinal property that is a strengthening
of
supercompactness.
A cardinal $\kappa$ is *$\alpha$-hypercompact* if and only if for
every ordinal $\beta < \alpha$ and for every cardinal
$\lambda\geq\kappa$, there exists a cardinal $\lambda’\geq\lambda$
and an elementary embedding $j:V\to M$ generated by a normal fine
ultrafilter
on $P_\kappa\lambda$ such that $\kappa$ is $\beta$-hypercompact in
$M$. $\kappa$ is *hypercompact* if and only if it is
$\beta$-hypercompact for every ordinal $\beta$.

Every cardinal is 0-hypercompact, and 1-hypercompactness is equivalent to supercompactness.

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