= Solution
A property $P$ of $\kappa$ <reflects below a cardinal>[reflects below] $\kappa$ when
$$
\{\mu<\kappa:P(\mu)\}
$$
is unbounded in $\kappa$. The standard <reflection by a beta-strong embedding>[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 to
$$
\exists\mu\,(\gamma<\mu<j(\kappa)\land P(\mu)).
$$
<Elementary embedding>[Elementarity] 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 $\kappa$ <set-theoretic absoluteness>[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 satisfies
$$
\exists\mu\,(\gamma<\mu<j(\kappa)\land\mu\text{ is strongly inaccessible}).
$$
Elementarity supplies a strongly inaccessible $\mu$ between $\gamma$ and $\kappa$ in $V_\lambda$. As $\gamma$ was arbitrary, the strongly inaccessible cardinals below $\kappa$ are unbounded.
Back to article page
