Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 116 / 2 / a / Solution

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.