Supercompact cardinal ($\kappa$)

A cardinal $\kappa$ is supercompact if for every cardinal $\lambda \ge \kappa$ there is an elementary embedding $j:V\to M$ into a transitive class with critical point of an elementary embedding $\kappa$, $j(\kappa )>\lambda$, and $M^{\lambda }\subseteq M$. This closure supplies every sufficiently small rank segment needed to transfer existential witnesses into $M$.

If $\kappa$ is a supercompact cardinal, each true ${\Sigma }_2$ statement, allowing parameters in $V_{\kappa }$, holds in $V_{\kappa }$. Write it as an existential witness followed by a ${\Pi }_1$ assertion. Choose a supercompact elementary embedding whose transitive target contains the witness in $V_{j(\kappa )}$. Downward absoluteness of the universal bounded-matrix assertion puts the witness in that rank segment, and elementarity reflects this to $V_{\kappa }$.