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$$\psi(\psi_I(0))$$ is a large countable ordinal. Michael Rathjen's ordinal collapsing function $$\psi$$ is used here along with $$I$$, the first inaccessible cardinal. $$\psi_I(0)$$ is 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 $$\vartheta(\Omega), \vartheta(\Omega_\Omega), \vartheta(\Omega_{\Omega_\Omega}), \vartheta(\Omega_{\Omega_{\Omega_\Omega}})\ldots$$

Ordinals, ordinal analysis and set theory