Countable limit of Extended Buchholz's function

\(\psi(\psi_I(0))\) is a large countable ordinal. Here \(I\) is the first inaccessible cardinal, and \(\psi\) is an unspecified or undefined ordinal collapsing function, which is at least different from Buchholz's OCF, Rathjen's standard OCF based on a weakly Mahlo cardinal, and its variant. \(\psi_I(0)\) is supposed to be the omega fixed point. It is the proof-theoretic ordinal of \(\Pi_1^1-\text{TR}_0\), a subsystem of second-order arithmetic.

As there is not currently a notation to define \(\psi(\psi_I(0))\) on the ordinal notations article, we define a simple notation to do this below:

(Note: this specific function - not the function used in the title - was added because it is believed that this function is easier to understand than the one used in the title.)

Let \(\Omega_0=1\), and if \(\alpha>0\), let \(\Omega_\alpha=\omega_\alpha\).

By convention, \(\Omega\) is short for \(\Omega_1\) and \(\vartheta\) is short for \(\vartheta_0\).

\(C_0(\alpha,\beta) = \beta\)

\(C_{n+1}(\alpha,\beta) = \{\gamma+\delta,\omega^\gamma,\Omega_\gamma,\vartheta_\gamma(\eta):\gamma,\delta,\eta\in C_n(\alpha,\beta);\eta<\alpha\}\)

\(C(\alpha,\beta) = \bigcup_{n<\omega}C_n(\alpha,\beta)\)

\(\vartheta_\nu(\alpha) = \min\{\beta:\Omega_\nu\leq\beta;C(\alpha,\beta)\cap\Omega_{\nu+1}\subseteq\beta\}\)

\(\psi(\psi_I(0))\) is the limit of the sequence \(\vartheta(\Omega), \vartheta(\Omega_\Omega), \vartheta(\Omega_{\Omega_\Omega}), \vartheta(\Omega_{\Omega_{\Omega_\Omega}})\ldots\)