\(\psi(\psi_I(0))\) 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

Basics: cardinal numbers · normal function · ordinal notation · ordinal numbers · fundamental sequence
Theories: Presburger arithmetic · Peano arithmetic · second-order arithmetic · ZFC
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\Gamma_0\) · \(\vartheta(\Omega^3)\) · \(\vartheta(\Omega^\omega)\) · \(\vartheta(\Omega^\Omega)\) · \(\vartheta(\varepsilon_{\Omega + 1})\) · \(\psi(\Omega_\omega)\) · \(\psi(\varepsilon_{\Omega_\omega + 1})\) · \(\psi(\psi_I(0))\)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^\text{CK}\) · \(\lambda,\zeta,\Sigma,\gamma\) · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal · more...

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