Difference between revisions of "User-blog:Julian Barathieu/Ordinal analyses"

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|-
 
|-
 
| $\psi_{\Omega_1}(\varepsilon_{\mathcal{M}+1})$
 
| $\psi_{\Omega_1}(\varepsilon_{\mathcal{M}+1})$
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| $\Delta^1_2-\text{CA}+\text{BI}+\text{(M)}$
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| $\text{KPM}$
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+
| [http://www.mathematik.uni-muenchen.de/~buchholz/articles/M1.pdf][https://pdfs.semanticscholar.org/a3ff/6ef9ca7db754139d541dbae579a73ee784db.pdf]
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| $\mathcal{M}$ is the least weakly Mahlo cardinal
 
|-
 
|-
 
| $\Psi^0_{\Omega_1}(\varepsilon_{\mathcal{K}+1})$
 
| $\Psi^0_{\Omega_1}(\varepsilon_{\mathcal{K}+1})$
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| $\text{KP}+(\Pi_3-\text{Reflection})$
 
| $\text{KP}+(\Pi_3-\text{Reflection})$
 
| [https://www1.maths.leeds.ac.uk/~rathjen/Ehab.pdf]
 
| [https://www1.maths.leeds.ac.uk/~rathjen/Ehab.pdf]
|
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| $\mathcal{K}$ is the least $\Pi^1_1$-indescribable cardinal
 
|-
 
|-
 
| $\Psi^{\varepsilon_{\Xi+1}}_\mathbb{X}$
 
| $\Psi^{\varepsilon_{\Xi+1}}_\mathbb{X}$
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| $\text{KP}+(\Pi_\omega-\text{Reflection})$
 
| $\text{KP}+(\Pi_\omega-\text{Reflection})$
 
| [https://d-nb.info/1017849250/34]
 
| [https://d-nb.info/1017849250/34]
 +
| $\Xi$ is the least $\Pi^2_0$-indescribable cardinal
 +
|-
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| $\Psi^{\varepsilon_{\Upsilon+1}}_\mathbb{H}$
 
|
 
|
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| $\text{Stability},\text{KPi}+\forall\alpha\exists\kappa$ $L_\kappa\prec_1 L_{\kappa+\alpha}$
 +
| [https://d-nb.info/1017849250/34][https://www1.maths.leeds.ac.uk/~rathjen/NSTAB.ps]
 +
| $\Upsilon$ is the least subtle cardinal
 
|-
 
|-
| $\Psi^{\varepsilon_{I+1}}_\mathbb{K}$
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| $\Psi^{\varepsilon_{\mathbf{\text{I}}+1}}_\mathbb{K}$
 
| $\Delta^1_2-\text{CA}+\text{BI}+\text{ parameter-free}$ $\Pi^1_2-\text{CA}$
 
| $\Delta^1_2-\text{CA}+\text{BI}+\text{ parameter-free}$ $\Pi^1_2-\text{CA}$
 
| $\text{KP}+\exists M(\text{Trans(M)}\land M\prec_1 V)$
 
| $\text{KP}+\exists M(\text{Trans(M)}\land M\prec_1 V)$
 
| [https://www1.maths.leeds.ac.uk/~rathjen/pime.pdf]
 
| [https://www1.maths.leeds.ac.uk/~rathjen/pime.pdf]
|
 
|-
 
| $\Psi^{\varepsilon_{\Upsilon+1}}_\mathbb{H}$
 
|
 
| $\text{Stability},\text{KPi}+\forall\alpha\exists\kappa$ $L_\kappa\prec_1 L_{\kappa+\alpha}$
 
| [https://d-nb.info/1017849250/34]
 
 
|
 
|
 
|}
 
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- $\text{KPM}$ (full analysis): [https://www1.maths.leeds.ac.uk/~rathjen/WELL.pdf], [https://www1.maths.leeds.ac.uk/~rathjen/KPM-Ordinal-Analysis.pdf]
 
- $\text{KPM}$ (full analysis): [https://www1.maths.leeds.ac.uk/~rathjen/WELL.pdf], [https://www1.maths.leeds.ac.uk/~rathjen/KPM-Ordinal-Analysis.pdf]
 
- $\Delta^1_2-\text{CA+BI }+$ parameter-free $\Pi^1_2-\text{CA}$ (full analysis): [https://www1.maths.leeds.ac.uk/~rathjen/pime.pdf]
 
 
- $\text{KP +}$ strong reflection principles (full analysis): [https://d-nb.info/1017849250/34]
 
 
- $\text{Stability}, \text{KP }+$ "for all $\alpha$, there is a $\alpha$-stable ordinal " (full analysis): [https://d-nb.info/1017849250/34], [https://www1.maths.leeds.ac.uk/~rathjen/NSTAB.ps]
 
 
- $\Pi^1_2-\text{CA}_0, \Pi^1_2-\text{CA+BI}, \Delta^1\_3-\text{CA}$: [https://www1.maths.leeds.ac.uk/~rathjen/BULF.pdf]
 
  
 
- $\text{KP + V=L +}$ "there is an uncountable regular cardinal" (full cut elimination): [https://arxiv.org/pdf/1801.10025.pdf]
 
- $\text{KP + V=L +}$ "there is an uncountable regular cardinal" (full cut elimination): [https://arxiv.org/pdf/1801.10025.pdf]
 
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
 

Revision as of 10:52, 12 February 2018

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]