# Supercompact cardinals and being witnessed by a structure of limited rank

Supercompact cardinals have reflection properties. If a cardinal with some property (say a 3-huge cardinal) that is witnessed by a structure of limited rank exists above a supercompact cardinal κ, then a cardinal with that property exists below κ.

What does "that is witnessed by a structure of limited rank" mean? What exactly is limited rank?

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What is meant here is a property $P$ such that a cardinal $\kappa$ satisfies $P$ iff there is an $\alpha>\kappa$ such that $V_\alpha\models\Psi(\kappa)$ for some appropriate $\Psi$ (that, of course, depends on $P$).
Supercompactness and strongness are not like that, since both require the existence of arbitrarily large measures with certain properties. On the other hand, $3$-hugeness, although in consistency strength much higher than supercompactness, is verified "locally" in the sense indicated in the previous paragraph.