Correct cardinal

A \(\Sigma_n\)-correct cardinal is a cardinal \(\kappa\) such that \(V_\kappa\) is a \(\Sigma_n\)-elementary substructure of \(V\), where \(\Sigma_n\) is from the Lévy hierarchy.^{[Citation needed]}

A regular cardinal \(\kappa\) is \(\Sigma_2\)-correct iff for every first-order formula \(\phi(x)\) and any \(x\in H_\kappa\), if \(\exists\alpha(H_\alpha\vDash\phi(x))\), then there is a \(\beta<\kappa\) such that \(H_\beta\vDash\phi(x)\).