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, and \(\psi_I(0)\) is supposed to be the omega fixed point. See the main article of the omega fixed point for the detail of the issue on the \(\psi\) function. 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 \(\vartheta(\Omega), \vartheta(\Omega_\Omega), \vartheta(\Omega_{\Omega_\Omega}), \vartheta(\Omega_{\Omega_{\Omega_\Omega}})\ldots\)