User blog comment:Scorcher007/Large countable ordinal notation up to Z2 and ZFC/@comment-11227630-20181122083848

The
 * G[G[G2[gg'1]]] - [ZFC-+"admissible and ωα exists"]

is wrong. Every uncountable cardinal is automatically admissible, recursively Mahlo, 1-stable, stable, ∑n-stable, and more to come. So asserting "admissible" to a cardinal is useless.

Instead, you may use a concept of predicate-free indescribables.

An ordinal \(\kappa\) is \(\Pi^m_n\)-indescribable on class A if for every \(\Pi_n\) formula \(\phi\), \(\forall R\in V_{\kappa+1}(\langle V_{\kappa+m},\in,R\rangle\models\phi\rightarrow\exists\alpha\in\kappa\cap A(\langle V_{\alpha+m},\in,R\cap V_\alpha\rangle\models\phi))\). This concept is very strong. \(\alpha\) is \(\Pi^0_0\)-indescribable on A iff it is \(\Pi^0_1\)-indescribable on A, iff \(\alpha=\sup(\alpha\cap A)\); but then an ordinal is \(\Pi^0_2\)-indescribable iff it is inaccessible.

"Predicate-free indescribable" is a weaker variant of that. An ordinal \(\kappa\) is predicate-free \(\Pi^m_n\)-indescribable on class A if for every \(\Pi_n\) formula \(\phi\), \(V_{\kappa+m}\models\phi\rightarrow\exists\alpha\in\kappa\cap A(V_{\alpha+m}\models\phi)\).

A related concept is \(\Sigma_n\)-extendible (or \(\Pi_n\)-extendible) cardinals. \(\kappa\) is \(\Sigma_n\)-extendible (respectively, \(\Pi_n\)-extendible) if \(\exists\alpha>0(V_\kappa\prec_{\Sigma_n}V_{\kappa+\alpha})\) (respectively, \(\exists\alpha>0(V_\kappa\prec_{\Pi_n}V_{\kappa+\alpha})\)). \(\Sigma_n\) or \(\Pi_n\)-extendible cardinals are smaller than the worldly cardinal and can be proved to exists in ZFC. And the target of a \(\Pi_n\)-extendible is a predicate-free \(\Pi^0_n\)-indescribable.

And DontDrinkH20 may know more about those cardinals.