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

What about these definitions of "indescribable property": So, for example, a 2-weakly compact cardinal \(\kappa\) is a weakly compact cardinal with \(\Pi_1^1\)-indescribable property over the set of weakly compact cardinals \(<\kappa\).
 * \(\kappa\) has "weakly compact property" over set A if for every \(\Pi_1\) first-order unary formula \(\phi\), \(\forall S\in V_\kappa(V_{\kappa+1}\models\phi(S)\rightarrow\exists\alpha\in A\cap\kappa(V_{\alpha+1}\models\phi(S\cap V_\alpha)))\). Here the \(\Pi_1^1\)-indescribability over set A is used.
 * \(\kappa\) has "\(\Pi_m^n\)-indescribable property" over set A if for every \(\Pi_m\) first-order unary formula \(\phi\), \(\forall S\in V_\kappa(V_{\kappa+n}\models\phi(S)\rightarrow\exists\alpha\in A\cap\kappa(V_{\alpha+n}\models\phi(S\cap V_\alpha)))\)

Is the least 2-weakly compact cardinal larger than the least "weakly compact cardinal below which weakly compact cardinals are stationary"?

Do these definitions fit the ordering in the "4"?