User-blog:Julian Barathieu/Ordinal analyses
From Cantor's Attic
Currently in construction. Do not edit.
This is entirely outdated and I maintain a table several times larger. Ask me for the full version on Discord (Julian#2338) or send a mail at julian (at) a-lec (dot) org.
EDIT: Read-only link for the current version https://www.overleaf.com/read/jwnpjbrskcbn
Contents
Arithmetical theories
Set theories
Ordinal-collapsing functions
Table of proof-theoretic ordinals and their corresponding theories
| Proof-theoretic ordinal | Arithmetical theories | Set theories | References | Notes |
|---|---|---|---|---|
| $\omega^\omega$ | $\text{RCA}_0,\text{WKL}_0$ | [1] | ||
| $\varepsilon_0$ | $\text{ACA}_0$ | $\text{KP}\setminus\text{\{Infinity\}}$ | [2][3] | First epsilon number |
| $\varepsilon_{\varepsilon_0}$ | $\text{ACA}$ | [4] | ||
| $\psi_{\Omega_1}(\Omega^{\omega})=\varphi(\omega,0)$ | $\Delta^1_1-\text{CR}$ | [5] | ||
| $\psi_{\Omega_1}(\Omega^{\varepsilon_0})=\varphi(\varepsilon_0,0)$ | $\Delta^1_1-\text{CA},\Sigma^1_1-\text{AC}$ | [6] | ||
| $\psi_{\Omega_1}(\Omega^\Omega)=\Gamma_0$ | $\text{ATR}_0,\Delta^1_1-\text{CA}+\text{BR}$ | $\text{KPi}^-,\text{CZF}^-+\exists\kappa(\kappa\text{ is inaccessible})$ | [7] | |
| $\theta(\delta_n,0)$ | $\text{ACA}_0+(\Pi^1_{n+1}-\text{BI})$ | $\text{KP}^-+(\Pi_{n+1}-\text{Foundation})$ | [8] | $\delta_1=\Omega^\omega,\delta_{n+1}=\Omega^{\delta_n}$ |
| $\theta(\eta_n,0)$ | $\text{ACA}+(\Pi^1_{n+1}-\text{BI})$ | $\text{KP}^-+\text{IND}+(\Pi_{n+1}-\text{Foundation})$ | [9] | $\eta_1=\Omega^{\varepsilon_0},\eta_{n+1}=\Omega^{\eta_n}$ |
| $\psi_{\Omega_1}(\varepsilon_{\Omega+1})$ | $\text{ACA}+\text{BI}$ | $\text{KP}$ | [10] | Bachmann-Howard ordinal |
| $\psi_{\Omega_1}(\Omega_\omega)$ | $\Pi^1_1-\text{CA}_0, \Delta^1_2-\text{CA}_0$ | [11] | ||
| $\psi_{\Omega_1}(\Omega_\omega\varepsilon_0)$ | $\Pi^1_1-\text{CA}$ | [12] | ||
| $\psi_{\Omega_1}(\varepsilon_{\Omega_\omega+1})$ | $\Pi^1_1-\text{CA}+\text{BI}$ | $\text{KPl}$ | [13] | Takeuti-Feferman-Buchholz ordinal |
| $\psi_{\Omega_1}(\Omega_{\omega^\omega})$ | $\Delta^1_2-\text{CR}$ | [14] | ||
| $\psi_{\Omega_1}(\Omega_{\varepsilon_0})$ | $\Delta^1_2-\text{CA}$ | [15] | ||
| $\psi_{\Omega_1}(\varepsilon_{\mathcal{I}+1})$ | $\Delta^1_2-\text{CA}+\text{BI}$ | $\text{KPi}$ | [16] | $\mathcal{I}$ is the least weakly inaccessible cardinal |
| $\psi_{\Omega_1}(\Omega_{\mathcal{I}_\omega})$ | $\text{KPh}$ | |||
| $\psi_{\Omega_1}(\varepsilon_{\mathcal{M}+1})$ | $\Delta^1_2-\text{CA}+\text{BI}+\text{(M)}$ | $\text{KPM}$ | [17][18] | $\mathcal{M}$ is the least weakly Mahlo cardinal |
| $\psi_{\Omega_1}(\Omega_{\mathcal{M}+\omega})$ | $\text{KPM}^+$ | |||
| $\Psi^0_{\Omega_1}(\varepsilon_{\mathcal{K}+1})$ | $\text{ACA}+\text{BI}+(\Pi^1_4-\beta\text{-model Reflection})$ | $\text{KP}+(\Pi_3-\text{Reflection})$ | [19] | $\mathcal{K}$ is the least $\Pi^1_1$-indescribable cardinal |
| $\Psi^{\varepsilon_{\Xi+1}}_\mathbb{X}$ | $\text{ACA}+\text{BI}+\beta\text{-model Reflection}$ | $\text{KP}+(\Pi_\omega-\text{Reflection})$ | [20] | $\Xi$ is the least $\Pi^2_0$-indescribable cardinal |
| $\Psi^{\varepsilon_{\Upsilon+1}}_\mathbb{H}$ | $\text{Stability},\text{KPi}+\forall\alpha\exists\kappa$ $L_\kappa\prec_1 L_{\kappa+\alpha}$ | [21][22] | $\Upsilon$ is the least subtle cardinal | |
| $\Psi^{\varepsilon_{\mathbf{\text{I}}+1}}_\mathbb{K}$ | $\Delta^1_2-\text{CA}+\text{BI}+\text{ parameter-free}$ $\Pi^1_2-\text{CA}$ | $\text{KP}+\exists M(\text{Trans(M)}\land M\prec_1 V)$ | [23] | See (*) |
Notes
(*) $\mathbf{\text{I}}$ is not the least weakly inaccessible $\mathcal{I}$. It has a somewhat technical definition; cf the reference given.
