# User-blog:Julian Barathieu/Ordinal analyses

Currently in construction. Do not edit.

## 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 | Ordinal name |
---|---|---|---|---|

$\varepsilon_0$ | $ACA_0$ | $KP\setminus Inf$ | Epsilon zero | |

$\psi_0(\varepsilon_{\Omega+1})$ | $ACA+BI$ | $KP$ | [1] | Bachmann-Howard ordinal |

$\psi_0(\Omega_\omega)$ | $\Pi^1_1-CA_0, \Delta^1_2-CA_0$ | [2] | ||

$\psi_0(\Omega_\omega\epsilon_0)$ | $\Pi^1_1-CA$ | [3] | ||

$\psi_0(\varepsilon_{\Omega_\omega+1})$ | $\Pi^1_1-CA+BI$ | [4] | Takeuti-Feferman-Buchholz ordinal | |

$\psi_0(\Omega_{\varepsilon_0})$ | $\Delta^1_2-CA$ | [5] |

- $\text{ACA}_0 + $ Kruskal's Tree Theorem (full analysis), $\text{ACA}_0 + \Pi^1_n-\text{BI}$ (full analysis):
https://www1.maths.leeds.ac.uk/~rathjen/KRUSKAL.neu.pdf

- Fragments of KP (restricted foundation) (full analysis): [6]

- $\text{KPi}$, $\Delta^1_2-\text{CA+BI}$ (full cut elimination): [7]

- $\text{KPM}$ (full analysis): [8], [9]

- $\Delta^1_2-\text{CA+BI +}$ parameter-free $\Pi^1_2-\text{CA}$ (full analysis): [10]

- $\text{KP +}$ strong reflection principles (full analysis): [11]

- $\text{Stability}, \text{KP +}$ "for all $\alpha$, there is a $\alpha$-stable ordinal " (full analysis): [12], [13]

- $\Pi^1_2-\text{CA}_0, \Pi^1_2-\text{CA+BI}, \Delta^1\_3-\text{CA}$: [14]

- $\text{KP + V=L +}$ "there is an uncountable regular cardinal" (full cut elimination): [15]

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