A cardinal \(\kappa\) is \(\Pi^n_m\)-indescribable if for every Πm proposition φ, and set A ⊆ Vκ with (Vκ+n, ∈, A) ⊧ φ there exists an α < κ with (Vα+n, ∈, A ∩ Vα) ⊧ φ. \(\Pi^1_1\)-indescribable cardinals are the same as weakly compact cardinals.