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. [math]\displaystyle{ \psi_\Omega(\alpha) }[/math] is a collapsing function such that [math]\displaystyle{ \psi_\Omega(\alpha) \lt \Omega }[/math], where [math]\displaystyle{ \Omega }[/math] 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, [math]\displaystyle{ \mathsf{KP\Pi_N} }[/math] represents Kripke-Platek set theory for a [math]\displaystyle{ \mathsf{\Pi_N} }[/math]-reflecting universe, [math]\displaystyle{ \mathbb{K}_N }[/math] is the smallest [math]\displaystyle{ \mathsf{\Pi_N} }[/math]-reflecting ordinal, [math]\displaystyle{ N }[/math] is a natural number [math]\displaystyle{ \gt 2 }[/math], and [math]\displaystyle{ \Omega_0 = 0 }[/math].

Definition

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

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