Arai's psi function

Arai's \(\psi\) function is an ordinal collapsing function introduced by Toshiyasu Arai based on first order reflection under an extension \(\textrm{ZFLK}_N\) of \(\textrm{ZFC}\).T. Arai, A simplified ordinal analysis of first-order reflection Arai also introduced a new simplified ordinal notation \(OT\) associated to Arai's \(\psi\). 
Using \(OT\), Arai verified several results in proof theory. The paper is devoted to show that for a \(\Sigma_1\)-sentence \(\theta\) such that \(\mathsf{KP\Pi_N \vdash \theta}\), there exists a finite \(n\) such that \(L_\alpha \models \theta\) for \(\alpha = \psi_\Omega(\Omega_n(\mathbb{K}_N + 1))\). Theorem 1.1 Arai also stated without an explicit proof that \(\textrm{KP}\Pi_N\) proves that each initial segment \(\{\alpha \in OT \colon \alpha < \psi_\Omega(\Omega_n(\mathbb{K}_N + 1))\}; n = 1, 2, \ldots\) is well-founded, where the comparison means the bound of the order type, Theorem 1.2 and concluded that \(\psi_{\Omega_1}(\varepsilon_{\mathbb{K}+1})\) is the proof-theoretic ordinal \(|\textrm{KP}\Pi_N|_{\Sigma_1^{\Omega}}\) of \(\textrm{KP}\Pi_N\) defined in terms of \(\Sigma_1\)-reflection, where \(\mathbb{K}\) is the least \(\Pi_N\)-reflecting ordinal and \(\textrm{KP}\Pi_N\) is Kripke–Platek set theory for the \(\Pi_N\)-reflecting universe. Theorem 1.3 
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== See also ==
Category:Ordinal collapsing functions
Category:Set theory