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
$\theta(\delta_n,0)$ $\text{ACA}_0+(\Pi^1_{n+1}-\text{BI})$ $\text{KP}^-+(\Pi_{n+1}-\text{Foundation})$ [2] $\delta_1=\Omega^\omega,\delta_{n+1}=\Omega^{\delta_n}$
$\theta(\eta_n,0)$ $\text{ACA}+(\Pi^1_{n+1}-BI)$ $\text{KP}^-+\text{IND}+(\Pi_{n+1}-\text{Foundation})$ [3] $\eta_1=\Omega^{\varepsilon_0},\eta_{n+1}=\Omega^{\eta_n}$
$\psi_0(\varepsilon_{\Omega+1})$ $\text{ACA+BI}$ $\text{KP}$ [4] Bachmann-Howard ordinal
$\psi_0(\Omega_\omega)$ $\Pi^1_1-\text{CA}_0, \Delta^1_2-\text{CA}_0$ [5]
$\psi_0(\Omega_\omega\varepsilon_0)$ $\Pi^1_1-\text{CA}$ [6]
$\psi_0(\varepsilon_{\Omega_\omega+1})$ $\Pi^1_1-\text{CA+BI}$ [7] Takeuti-Feferman-Buchholz ordinal
$\psi_0(\Omega_{\varepsilon_0})$ $\Delta^1_2-\text{CA}$ [8]

- $\text{KPi}$, $\Delta^1_2-\text{CA+BI}$ (full cut elimination): [9]

- $\text{KPM}$ (full analysis): [10], [11]

- $\Delta^1_2-\text{CA+BI +}$ parameter-free $\Pi^1_2-\text{CA}$ (full analysis): [12]

- $\text{KP +}$ strong reflection principles (full analysis): [13]

- $\text{Stability}, \text{KP +}$ "for all $\alpha$, there is a $\alpha$-stable ordinal " (full analysis): [14], [15]

- $\Pi^1_2-\text{CA}_0, \Pi^1_2-\text{CA+BI}, \Delta^1\_3-\text{CA}$: [16]

- $\text{KP + V=L +}$ "there is an uncountable regular cardinal" (full cut elimination): [17]

https://www1.maths.leeds.ac.uk/~rathjen/Sepp-chiemsee.pdf - Pi^1\_1-CA0, ID<w, Pi^1\_1-CA - Pi^1\_1-CA+BI, IDw, Delta^1\_2-CR - Delta^1\_2-CA, Delta^1\_2-CA+BI = KPi