User-blog:Julian Barathieu/Ordinal analyses

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Currently in construction. Do not edit.

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.