User blog:Googleaarex/A new, really strong OCF definition?

Definition
Nothing going to beat my OCF now, here is the definition.

\(D[0]\) is least cardinality and \(D[\alpha+1]\) is greater cardinality than \(D[\alpha]\). Also, \(D[\alpha[D_0]]\) is similar to weakly inaccessible cardinals in OCFs, which:
 * \(\psi_{D[\alpha[D_0]]}(0)\) = \(\beta\) such that \(\beta\) = \(D[\alpha[\beta]]\)
 * \(\psi_{D[\alpha[D_0]]}(\beta+1)\) = \(\gamma\) such that \(\gamma\) = \(D[\alpha[\psi_{D[\alpha[D_0]]}(\beta)+\gamma]]\)

Then a new definition happen to prove it WAY stronger:
 * D[\alpha[\beta[D_{\gamma}]]] = D[\alpha[\beta[D[\alpha[D_{\gamma+1}]]]]], where \(\beta\) = \(min\{\beta|\beta < D_{\gamma+1}\}\)
 * D[\alpha[D_{\beta+1}]] is a \(D_{\beta}\) cardinal such that it works as in D[\alpha[\text{_}]].

I name the notation \(D[\alpha]\) as Dropper Ordinal Notation.

Rules
Now here is OCF expansion rules: Work in progress!
 * \(\psi_\Omega(0)\) = \(\alpha\) such that \(\alpha\) = \(\omega^{\alpha}\)
 * \(\psi_{\Omega_{\alpha+1}}(0)\) = \(\beta\) such that \(\beta\) = \(\Omega_{\alpha+1}^{\beta}\)
 * \(\psi_{\Omega_{\alpha+1}}(\beta+1)\) = \(\gamma\) such that \(\gamma\) = \(\psi_{\Omega_{\alpha+1}}(\beta)^{\gamma}\)
 * \(\psi_{\Omega_{\alpha+1}}(\beta)[\gamma]\) = \(\psi_{\Omega_{\alpha+1}}(\beta[\gamma])\)
 * \(\psi_\Omega\)
 * \(\psi_{\alpha}(\beta)\) = \(\alpha\)