User blog comment:Hyp cos/TON, stable ordinals, and my array notation/@comment-31580368-20180927133742/@comment-11227630-20180927142336

I'm not very sure about this, but I guess there are \(\omega\) levels of largeness between 1-stable and 2-stable. Ordinal \(\alpha\) is \(\beta\)-\(\Pi_n\)-reflecting if for every \(\Pi_n\) formula \(\phi\), \(L_{\alpha+\beta}\models\phi\rightarrow\exists\gamma<\alpha(L_{\gamma+\beta}\models\phi)\).

This is similar to \(\beta\)-\(\Pi_n^1\)-indescribable. \(\alpha\) is \(\beta\)-\(\Pi_n^1\)-indescribable if for every \(\Pi_n^1\) formula \(\phi\), \(V_{\alpha+\beta}\models\phi\rightarrow\exists\gamma<\alpha(V_{\gamma+\beta}\models\phi)\).