Arai ψ function

Arai's ψ function is an ordinal collapsing function invented by mathematician Toshiyasu Arai (husband of Noriko H. Arai) in his paper: A simplified ordinal analysis of first-order reflection. ${\psi }_{\mathrm{\Omega }}(\alpha )$ is a collapsing function such that ${\psi }_{\mathrm{\Omega }}(\alpha )<\mathrm{\Omega }$, where $\mathrm{\Omega }$ 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, $\mathsf{K}\mathsf{P}{\mathrm{\Pi }}_{\mathsf{N}}$ represents Kripke-Platek set theory for a ${\mathrm{\Pi }}_{\mathsf{N}}$-reflecting universe, ${\mathbb{𝕂}}_N$ is the smallest ${\mathrm{\Pi }}_{\mathsf{N}}$-reflecting ordinal, $N$ is a natural number $>2$, and ${\mathrm{\Omega }}_0=0$.

Definition

Suppose $\mathsf{K}\mathsf{P}{\mathrm{\Pi }}_{\mathsf{N}}\mathsf{⊢}\mathsf{\theta }$ for a ${\mathrm{\Sigma }}_{\mathsf{1}}$ ($\mathrm{\Omega }$)-sentence $\mathsf{\theta }$. Then, there exists a finite $n$ such that for $\alpha ={\psi }_{\mathrm{\Omega }}({\mathrm{\Omega }}_n({\mathbb{𝕂}}_N+1))$, $L_{\alpha }\models \theta$. It can also be proven that $\mathsf{K}\mathsf{P}{\mathrm{\Pi }}_{\mathsf{N}}$ proves that each initial segment $\{\alpha \in OT:\alpha <{\psi }_{\mathrm{\Omega }}({\mathrm{\Omega }}_n({\mathbb{𝕂}}_N+1))\};n=1,2,...$ is well-founded, and therefore, the proof-theoretic ordinal of ${\psi }_{\mathrm{\Omega }}({\epsilon }_{{\mathbb{𝕂}}_N+1})$ is the proof-theoretic ordinal of $\mathsf{K}\mathsf{P}{\mathrm{\Pi }}_{\mathsf{N}}$. Using this, ${\psi }_{\mathrm{\Omega }}({\epsilon }_{{\mathbb{𝕂}}_N+1})=min(\{\alpha \le \mathrm{\Omega }|\mathrm{\forall }\theta \in {\mathrm{\Sigma }}_1(\mathsf{K}\mathsf{P}{\mathrm{\Pi }}_{\mathsf{N}}⊢{\theta }^{L_{\mathrm{\Omega }}}\to L_{\alpha }\models \theta )\})$. One can then make the following conversions:

• ${\psi }_{\mathrm{\Omega }}(A)={\psi }_0(\mathrm{\Omega })=|\mathsf{P}\mathsf{A}|=\phi (1,0)$, where $A$ is the least admissible ordinal, $\mathsf{P}\mathsf{A}$ is Peano arithmetic and $\phi$ is the Veblen hierarchy.
• ${\psi }_{\mathrm{\Omega }}({\epsilon }_{A+1})={\psi }_0({\epsilon }_{\mathrm{\Omega }+1})=|\mathsf{K}\mathsf{P}|=\mathsf{B}\mathsf{H}\mathsf{O}$, where $A$ is the least admissible ordinal, $\mathsf{K}\mathsf{P}$ is Kripke-Platek set theory and $\mathsf{B}\mathsf{H}\mathsf{O}$ is the Bachmann-Howard ordinal.
• ${\psi }_{\mathrm{\Omega }}(I)={\psi }_0({\mathrm{\Omega }}_{\omega })=|{\mathrm{\Pi }}_{\mathsf{1}}^{\mathsf{1}}\mathsf{-}\mathsf{C}{\mathsf{A}}_{\mathsf{0}}|=\mathsf{B}\mathsf{O}$, where $I$ is the least recursively inaccessible ordinal and $\mathsf{B}\mathsf{O}$ is Buchholz's ordinal.
• ${\psi }_{\mathrm{\Omega }}({\epsilon }_{I+1})={\psi }_0({\epsilon }_{{\mathrm{\Omega }}_{\omega }+1})=|\mathsf{K}\mathsf{P}\mathsf{I}|=\mathsf{T}\mathsf{F}\mathsf{B}\mathsf{O}$, where $I$ is the least recursively inaccessible ordinal, $\mathsf{K}\mathsf{P}\mathsf{I}$ is Kripke-Platek set theory with a recursively inaccessible universe and $\mathsf{T}\mathsf{F}\mathsf{B}\mathsf{O}$ 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.