| rdfs:comment
| - \(\psi(\psi_I(0))\) 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-theoritic ordinal of \(\Pi_1^1- ext{TR}_0\), a susbystem 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: Let \(\Omega_0=1\), and if \(\alpha>0\), let \(\Omega_\alpha=\omega_\alpha\). \(C_0(\alpha,\beta) = \beta\) \(C(\alpha,\beta) = \bigcup_{n<\omega}C_n(\alpha,\beta)\)
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| abstract
| - \(\psi(\psi_I(0))\) 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-theoritic ordinal of \(\Pi_1^1- ext{TR}_0\), a susbystem 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 \(\vartheta_0\). \(C_0(\alpha,\beta) = \beta\) \(C_{n+1}(\alpha,\beta) = \{\gamma+\delta,\omega^\gamma,\Omega_\gamma,\vartheta_\gamma(\eta):\gamma,\delta,\eta\in C_n(\alpha,\beta);\eta<\alpha\}\) \(C(\alpha,\beta) = \bigcup_{n<\omega}C_n(\alpha,\beta)\) \(\vartheta_
u(\alpha) = \min\{\beta:\Omega_
u\leq\beta;C(\alpha,\beta)\cap\Omega_{
u+1}\subseteq\beta\}\) \(\psi(\psi_I(0))\) 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 numbersTheories: Presburger arithmetic · Peano arithmetic · second-order arithmetic · ZFCCountable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\Gamma_0\) · \(\vartheta(\Omega^3)\) · \(\vartheta(\Omega^\omega)\) · \(\vartheta(\Omega^\Omega)\) · \(\vartheta(\varepsilon_{\Omega + 1})\) · \(\psi(\Omega_\omega)\) · \(\psi(\varepsilon_{\Omega_\omega + 1})\) · \(\psi(\psi_I(0))\) · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^ ext{CK}\) · \(\lambda,\zeta,\Sigma,\gamma\)Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchyUncountable 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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