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A property $P$ of $\kappa$ reflects below $\kappa$ whenis unbounded in $\kappa$. The standard elementary-embedding reflection argument starts with any $\gamma <\kappa$. Since $j(\gamma )=\gamma$ and $j(\kappa )>\kappa$, the target model can use $\kappa$ itself as a witness toElementarity then gives a witness $\mu$ with $\gamma <\mu <\kappa$ in the domain.

For a concrete example, the property of being a strongly inaccessible cardinal reflects below $\kappa$. A measurable cardinal is strongly inaccessible, and the assumed inclusion $V_{\kappa +1}\subseteq M$ makes this assertion about $\kappa$ absolute between $V_{\lambda }$ and $M$: both models have all subsets of every ordinal below $\kappa$. Thus $M$ sees that $\kappa$ is strongly inaccessible. Given $\gamma <\kappa$, it therefore satisfiesElementarity supplies a strongly inaccessible $\mu$ between $\gamma$ and $\kappa$ in $V_{\lambda }$. As $\gamma$ was arbitrary, the strongly inaccessible cardinals below $\kappa$ are unbounded.

Let $\lambda$ be inaccessible and let $j:V_{\lambda }\to M$ be an elementary embedding into a transitive set with critical point $\kappa$. It is $\beta$-strong whenA formula $\Phi (x,\kappa )$ is a beta-stable cardinal property when it is absolute between the universe and every transitive set containing $V_{\kappa +\beta }$.

To say that the embedding reflects such a property means that whenever $\Phi (\kappa )$ holds,is unbounded in $\kappa$: for every $\gamma <\kappa$ there is such a $\mu$ with $\gamma <\mu <\kappa$. Indeed, stability gives $M\models \Phi (\kappa )$, so $M$ sees the witness $\kappa$ between $\gamma$ and $j(\kappa )$. Elementarity reflects a witness between $\gamma$ and $\kappa$. This is reflection by a beta-strong embedding.