Beta-strong elementary embedding ($\beta$-strong)

For inaccessible $\lambda$, a map $j:V_{\lambda }\to M$ with critical point $\kappa$ is $\beta$-strong when $M$ is transitive and $V_{\kappa +\beta }\subseteq M$.

A cardinal $\kappa <\lambda$ is 1-strong when it is the critical point of an elementary embedding $j:V_{\lambda }\to M$ into a transitive model satisfying $V_{\kappa +1}\subseteq M$.

Fix an ordinal $\beta$ and a one-variable first-order formula $\Phi (v)$ expressing a cardinal property. In this context the property is $\beta$-stable at $\kappa$ when its truth at $\kappa$ is absolute between the ambient universe and every relevant transitive target $M$ containing $V_{\kappa +\beta }$:Relevant targets must satisfy the background set-theoretic axioms used in the argument. If the formula uses additional parameters, they must be specified and kept fixed by the elementary embedding in a reflection application.

For a beta-strong elementary embedding, this absoluteness lets the target recognize the property at the old critical point $\kappa$. The inequality $\kappa <j(\kappa )$ then supplies a witness below $j(\kappa )$, and elementarity reflects witnesses below $\kappa$. Writing an unexplained second parameter in $\Phi (x,\kappa )$ would conceal that the embedding may move that parameter as well.

If a $\beta$-strong embedding has critical point $\kappa$ and the $\beta$-stable property $\Phi (\kappa )$ holds, then $\{\mu <\kappa :\Phi (\mu )\}$ is unbounded in $\kappa$. For each $\gamma <\kappa$, the target sees $\kappa$ as a witness between $\gamma$ and $j(\kappa )$; elementarity reflects a witness between $\gamma$ and $\kappa$.

A property $P$ reflects below a cardinal $\kappa$ when $\{\mu <\kappa :P(\mu )\}$ is unbounded in $\kappa$. For an elementary embedding with critical point $\kappa$, one often proves this by using $\kappa$ as a witness below $j(\kappa )$ in the target and then invoking elementarity.