# Arai psi function

In mathematics, Arai's ψ function is an ordinal collapsing function introduced by Toshiyasu Arai (husband of Noriko H. Arai) in his paper: A simplified ordinal analysis of first-order reflection. $\displaystyle{ \psi_\Omega(\alpha) }$ is a collapsing function such that $\displaystyle{ \psi_\Omega(\alpha) \lt \Omega }$, where $\displaystyle{ \Omega }$ represents the first uncountable ordinal (it can be replaced by the Church–Kleene ordinal at the cost of extra technical difficulty). Throughout the course of this article, $\displaystyle{ \mathsf{KP\Pi_N} }$ represents Kripke–Platek set theory for a $\displaystyle{ \mathsf{\Pi_N} }$-reflecting universe, $\displaystyle{ \mathbb{K}_N }$ is the smallest $\displaystyle{ \mathsf{\Pi_N} }$-reflecting ordinal, $\displaystyle{ N }$ is a natural number $\displaystyle{ \gt 2 }$, and $\displaystyle{ \Omega_0 = 0 }$.

## Definition

Suppose $\displaystyle{ \mathsf{KP\Pi_N \vdash \theta} }$ for a $\displaystyle{ \mathsf{\Sigma_1} }$ ($\displaystyle{ \Omega }$)-sentence $\displaystyle{ \mathsf{\theta} }$. Then, there exists a finite $\displaystyle{ n }$ such that for $\displaystyle{ \alpha = \psi_\Omega(\Omega_n(\mathbb{K}_N + 1)) }$, $\displaystyle{ L_\alpha \models \theta }$. It can also be proven that $\displaystyle{ \mathsf{KP\Pi_N} }$ proves that each initial segment $\displaystyle{ \{\alpha \in OT: \alpha \lt \psi_\Omega(\Omega_n(\mathbb{K}_N + 1))\}; n = 1, 2, \ldots }$ is well-founded, and therefore, the proof-theoretic ordinal of $\displaystyle{ \psi_\Omega(\varepsilon_{\mathbb{K}_N+1}) }$ is the proof-theoretic ordinal of $\displaystyle{ \mathsf{KP\Pi_N} }$. Using this, $\displaystyle{ \psi_\Omega(\varepsilon_{\mathbb{K}_N+1}) = \min(\{\alpha \leq \Omega \mid \forall \theta \in \Sigma_1(\mathsf{KP\Pi_N} \vdash \theta^{L_\Omega} \rightarrow L_\alpha \models \theta)\}) }$. One can then make the following conversions:

• $\displaystyle{ \psi_\Omega(A) = \psi_0(\Omega) = |\mathsf{PA}| = \varphi(1, 0) }$, where $\displaystyle{ A }$ is the least admissible ordinal, $\displaystyle{ \mathsf{PA} }$ is Peano arithmetic and $\displaystyle{ \varphi }$ is the Veblen hierarchy.
• $\displaystyle{ \psi_\Omega(\varepsilon_{A +1}) = \psi_0(\varepsilon_{\Omega + 1}) = |\mathsf{KP}| = \mathsf{BHO} }$, where $\displaystyle{ A }$ is the least admissible ordinal, $\displaystyle{ \mathsf{KP} }$ is Kripke–Platek set theory and $\displaystyle{ \mathsf{BHO} }$ is the Bachmann–Howard ordinal.
• $\displaystyle{ \psi_\Omega(I) = \psi_0(\Omega_\omega) = |\mathsf{\Pi^1_1-CA_0}| = \mathsf{BO} }$, where $\displaystyle{ I }$ is the least recursively inaccessible ordinal and $\displaystyle{ \mathsf{BO} }$ is Buchholz's ordinal.
• $\displaystyle{ \psi_\Omega(\varepsilon_{I +1}) = \psi_0(\varepsilon_{\Omega_\omega + 1}) = |\mathsf{KPI}| = \mathsf{TFBO} }$, where $\displaystyle{ I }$ is the least recursively inaccessible ordinal, $\displaystyle{ \mathsf{KPI} }$ is Kripke–Platek set theory with a recursively inaccessible universe and $\displaystyle{ \mathsf{TFBO} }$ is the Takeuti–Feferman–Buchholz ordinal.

## References

• Arai, Toshiyasu (September 2020). "A simplified ordinal analysis of first-order reflection". The Journal of Symbolic Logic 85 (3): 1163–1185. doi:10.1017/jsl.2020.23.