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. $\psi _{\Omega }(\alpha )$ is a collapsing function such that $\psi _{\Omega }(\alpha )<\Omega$, where $\Omega$ 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, ${\mathsf {KP\Pi _{N}}}$ represents Kripke–Platek set theory for a ${\mathsf {\Pi _{N}}}$-reflecting universe, $\mathbb {K} _{N}$ is the smallest ${\mathsf {\Pi _{N}}}$-reflecting ordinal, $N$ is a natural number $>2$, and $\Omega _{0}=0$.

Definition

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

• $\psi _{\Omega }(A)=\psi _{0}(\Omega )=|{\mathsf {PA}}|=\varphi (1,0)$, where $A$ is the least admissible ordinal, ${\mathsf {PA}}$ is Peano arithmetic and $\varphi$ is the Veblen hierarchy.
• $\psi _{\Omega }(\varepsilon _{A+1})=\psi _{0}(\varepsilon _{\Omega +1})=|{\mathsf {KP}}|={\mathsf {BHO}}$, where $A$ is the least admissible ordinal, ${\mathsf {KP}}$ is Kripke–Platek set theory and ${\mathsf {BHO}}$ is the Bachmann–Howard ordinal.
• $\psi _{\Omega }(I)=\psi _{0}(\Omega _{\omega })=|{\mathsf {\Pi _{1}^{1}-CA_{0}}}|={\mathsf {BO}}$, where $I$ is the least recursively inaccessible ordinal and ${\mathsf {BO}}$ is Buchholz's ordinal.
• $\psi _{\Omega }(\varepsilon _{I+1})=\psi _{0}(\varepsilon _{\Omega _{\omega }+1})=|{\mathsf {KPI}}|={\mathsf {TFBO}}$, where $I$ is the least recursively inaccessible ordinal, ${\mathsf {KPI}}$ is Kripke–Platek set theory with a recursively inaccessible universe and ${\mathsf {TFBO}}$ 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. arXiv:1907.07611. doi:10.1017/jsl.2020.23.