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
$\varepsilon_0$ $\text{ACA}_0$ $\text{KP}\setminus\text{\{Infinity\}}$ [1] First epsilon number
$\Gamma_0$ $\text{ATR}_0$ $\text{KPi}^-,\text{CZF}^-+\exists\kappa(\kappa\text{ is inaccessible})$ [2]
$\theta(\delta_n,0)$ $\text{ACA}_0+(\Pi^1_{n+1}-\text{BI})$ $\text{KP}^-+(\Pi_{n+1}-\text{Foundation})$ [3] $\delta_1=\Omega^\omega,\delta_{n+1}=\Omega^{\delta_n}$
$\theta(\eta_n,0)$ $\text{ACA}+(\Pi^1_{n+1}-\text{BI})$ $\text{KP}^-+\text{IND}+(\Pi_{n+1}-\text{Foundation})$ [4] $\eta_1=\Omega^{\varepsilon_0},\eta_{n+1}=\Omega^{\eta_n}$
$\psi_{\Omega_1}(\varepsilon_{\Omega+1})$ $\text{ACA}+\text{BI}$ $\text{KP}$ [5] Bachmann-Howard ordinal
$\psi_{\Omega_1}(\Omega_\omega)$ $\Pi^1_1-\text{CA}_0, \Delta^1_2-\text{CA}_0$ [6]
$\psi_{\Omega_1}(\Omega_\omega\varepsilon_0)$ $\Pi^1_1-\text{CA}$ [7]
$\psi_{\Omega_1}(\varepsilon_{\Omega_\omega+1})$ $\Pi^1_1-\text{CA}+\text{BI}$ [8] Takeuti-Feferman-Buchholz ordinal
$\psi_{\Omega_1}(\Omega_{\varepsilon_0})$ $\Delta^1_2-\text{CA}$ [9]
$\psi_{\Omega_1}(\varepsilon_{\mathcal{M}+1})$ $\Delta^1_2-\text{CA}+\text{BI}+\text{(M)}$ $\text{KPM}$ [10][11] $\mathcal{M}$ is the least weakly Mahlo cardinal
$\Psi^0_{\Omega_1}(\varepsilon_{\mathcal{K}+1})$ $\text{ACA}+\text{BI}+(\Pi^1_4-\beta\text{-model Reflection})$ $\text{KP}+(\Pi_3-\text{Reflection})$ [12] $\mathcal{K}$ is the least $\Pi^1_1$-indescribable cardinal
$\Psi^{\varepsilon_{\Xi+1}}_\mathbb{X}$ $\text{ACA}+\text{BI}+\beta\text{-model Reflection}$ $\text{KP}+(\Pi_\omega-\text{Reflection})$ [13] $\Xi$ is the least $\Pi^2_0$-indescribable cardinal
$\Psi^{\varepsilon_{\Upsilon+1}}_\mathbb{H}$ $\text{Stability},\text{KPi}+\forall\alpha\exists\kappa$ $L_\kappa\prec_1 L_{\kappa+\alpha}$ [14][15] $\Upsilon$ is the least subtle cardinal
$\Psi^{\varepsilon_{\mathbf{\text{I}}+1}}_\mathbb{K}$ $\Delta^1_2-\text{CA}+\text{BI}+\text{ parameter-free}$ $\Pi^1_2-\text{CA}$ $\text{KP}+\exists M(\text{Trans(M)}\land M\prec_1 V)$ [16]

- $\text{KPi}$, $\Delta^1_2-\text{CA}+\text{BI}$ (full cut elimination): [17]

- $\text{KPM}$ (full analysis): [18], [19]

- $\text{KP + V=L +}$ "there is an uncountable regular cardinal" (full cut elimination): [20]