Difference between revisions of "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 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