User blog comment:Hyp cos/Question about weak compactness/@comment-11227630-20180913145946/@comment-35470197-20180914005917

I think that it is easier to use the satisfaction at \(V_{\kappa}\).

Namely, an inaccessible cardinal \(\kappa\) has "\(\Pi_m^n\)-indescriable property over \(A\)" if for any \(\Pi_m\)-formula \(\phi\), the condition \begin{eqnaray*} S \in V_{\kappa+1}(((V_{\kappa},S) \models \phi) \to (\exists \alpha \in A \cap \kappa((V_{\alpha},S \cap V_{\alpha}) \models \phi))) \end{eqnarray*} holds.

In this sense, the least 2-weakly compact cardinal below which the set of weakly compact cardinals is stationary is "2-weakly compact".