Difference between revisions of "User-blog:Julian Barathieu/Ordinal analyses"
From Cantor's Attic
| Line 16: | Line 16: | ||
! References | ! References | ||
! Notes | ! Notes | ||
| + | |- | ||
| + | | $\omega^\omega$ | ||
| + | | $\text{RCA}_0,\text{WKL}_0$ | ||
| + | | | ||
| + | | [https://www1.maths.leeds.ac.uk/~rathjen/Sepp-chiemsee.pdf] | ||
| + | | | ||
|- | |- | ||
| $\varepsilon_0$ | | $\varepsilon_0$ | ||
| $\text{ACA}_0$ | | $\text{ACA}_0$ | ||
| $\text{KP}\setminus\text{\{Infinity\}}$ | | $\text{KP}\setminus\text{\{Infinity\}}$ | ||
| − | | [https://www1.maths.leeds.ac.uk/~rathjen/Sepp-chiemsee.pdf] | + | | [https://www1.maths.leeds.ac.uk/~rathjen/Sepp-chiemsee.pdf][http://www.mathematik.uni-muenchen.de/~buchholz/articles/BuchholzGentzCent1.pdf] |
| First epsilon number | | First epsilon number | ||
|- | |- | ||
| − | | $\Gamma_0$ | + | | $\varepsilon_{\varepsilon_0}$ |
| − | | $\text{ATR}_0$ | + | | $\text{ACA}$ |
| + | | | ||
| + | | [https://www1.maths.leeds.ac.uk/~rathjen/Sepp-chiemsee.pdf] | ||
| + | | | ||
| + | |- | ||
| + | | $\psi_{\Omega_1}(\Omega^{\omega})=\varphi(\omega,0)$ | ||
| + | | $\Delta^1_1-\text{CR}$ | ||
| + | | | ||
| + | | [https://www1.maths.leeds.ac.uk/~rathjen/Sepp-chiemsee.pdf] | ||
| + | | | ||
| + | |- | ||
| + | | $\psi_{\Omega_1}(\Omega^{\varepsilon_0})=\varphi(\varepsilon_0,0)$ | ||
| + | | $\Delta^1_1-\text{CA},\Sigma^1_1-\text{AC}$ | ||
| + | | | ||
| + | | [https://www1.maths.leeds.ac.uk/~rathjen/Sepp-chiemsee.pdf] | ||
| + | | | ||
| + | |- | ||
| + | | $\psi_{\Omega_1}(\Omega^\Omega)=\Gamma_0$ | ||
| + | | $\text{ATR}_0,\Delta^1_1-\text{CA}+\text{BR}$ | ||
| $\text{KPi}^-,\text{CZF}^-+\exists\kappa(\kappa\text{ is inaccessible})$ | | $\text{KPi}^-,\text{CZF}^-+\exists\kappa(\kappa\text{ is inaccessible})$ | ||
| [https://www1.maths.leeds.ac.uk/~rathjen/Sepp-chiemsee.pdf] | | [https://www1.maths.leeds.ac.uk/~rathjen/Sepp-chiemsee.pdf] | ||
| Line 61: | Line 85: | ||
| $\psi_{\Omega_1}(\varepsilon_{\Omega_\omega+1})$ | | $\psi_{\Omega_1}(\varepsilon_{\Omega_\omega+1})$ | ||
| $\Pi^1_1-\text{CA}+\text{BI}$ | | $\Pi^1_1-\text{CA}+\text{BI}$ | ||
| − | | | + | | $\text{KPl}$ |
| [https://www1.maths.leeds.ac.uk/~rathjen/Sepp-chiemsee.pdf] | | [https://www1.maths.leeds.ac.uk/~rathjen/Sepp-chiemsee.pdf] | ||
| Takeuti-Feferman-Buchholz ordinal | | Takeuti-Feferman-Buchholz ordinal | ||
| + | |- | ||
| + | | $\psi_{\Omega_1}(\Omega_{\omega^\omega})$ | ||
| + | | $\Delta^1_2-\text{CR}$ | ||
| + | | | ||
| + | | [https://www1.maths.leeds.ac.uk/~rathjen/Sepp-chiemsee.pdf] | ||
| + | | | ||
|- | |- | ||
| $\psi_{\Omega_1}(\Omega_{\varepsilon_0})$ | | $\psi_{\Omega_1}(\Omega_{\varepsilon_0})$ | ||
| Line 69: | Line 99: | ||
| | | | ||
| [https://www1.maths.leeds.ac.uk/~rathjen/Sepp-chiemsee.pdf] | | [https://www1.maths.leeds.ac.uk/~rathjen/Sepp-chiemsee.pdf] | ||
| + | | | ||
| + | |- | ||
| + | | $\psi_{\Omega_1}(\varepsilon_{\mathcal{I}+1})$ | ||
| + | | $\Delta^1_2-\text{CA}+\text{BI}$ | ||
| + | | $\text{KPi}$ | ||
| + | | [https://www1.maths.leeds.ac.uk/~rathjen/Sepp-chiemsee.pdf] | ||
| + | | $\mathcal{I}$ is the least weakly inaccessible cardinal | ||
| + | |- | ||
| + | | $\psi_{\Omega_1}(\Omega_{\mathcal{I}_\omega})$ | ||
| + | | | ||
| + | | $\text{KPh}$ | ||
| + | | | ||
| | | | ||
|- | |- | ||
| Line 76: | Line 118: | ||
| [http://www.mathematik.uni-muenchen.de/~buchholz/articles/M1.pdf][https://pdfs.semanticscholar.org/a3ff/6ef9ca7db754139d541dbae579a73ee784db.pdf] | | [http://www.mathematik.uni-muenchen.de/~buchholz/articles/M1.pdf][https://pdfs.semanticscholar.org/a3ff/6ef9ca7db754139d541dbae579a73ee784db.pdf] | ||
| $\mathcal{M}$ is the least weakly Mahlo cardinal | | $\mathcal{M}$ is the least weakly Mahlo cardinal | ||
| + | |- | ||
| + | | $\psi_{\Omega_1}(\Omega_{\mathcal{M}+\omega})$ | ||
| + | | | ||
| + | | $\text{KPM}^+$ | ||
| + | | | ||
| + | | | ||
|- | |- | ||
| $\Psi^0_{\Omega_1}(\varepsilon_{\mathcal{K}+1})$ | | $\Psi^0_{\Omega_1}(\varepsilon_{\mathcal{K}+1})$ | ||
| Line 99: | Line 147: | ||
| $\text{KP}+\exists M(\text{Trans(M)}\land M\prec_1 V)$ | | $\text{KP}+\exists M(\text{Trans(M)}\land M\prec_1 V)$ | ||
| [https://www1.maths.leeds.ac.uk/~rathjen/pime.pdf] | | [https://www1.maths.leeds.ac.uk/~rathjen/pime.pdf] | ||
| − | | | + | | See (*) |
|} | |} | ||
| − | + | == Notes == | |
| − | + | (*) $\mathbf{\text{I}}$ is not the least weakly inaccessible $\mathcal{I}$. It has a somewhat technical definition; cf the reference given. | |
| − | + | <!-- phi(A,B,C,D) = psi() --> | |
| − | + | ||
| − | + | ||
Revision as of 12:59, 12 February 2018
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 | Notes |
|---|---|---|---|---|
| $\omega^\omega$ | $\text{RCA}_0,\text{WKL}_0$ | [1] | ||
| $\varepsilon_0$ | $\text{ACA}_0$ | $\text{KP}\setminus\text{\{Infinity\}}$ | [2][3] | First epsilon number |
| $\varepsilon_{\varepsilon_0}$ | $\text{ACA}$ | [4] | ||
| $\psi_{\Omega_1}(\Omega^{\omega})=\varphi(\omega,0)$ | $\Delta^1_1-\text{CR}$ | [5] | ||
| $\psi_{\Omega_1}(\Omega^{\varepsilon_0})=\varphi(\varepsilon_0,0)$ | $\Delta^1_1-\text{CA},\Sigma^1_1-\text{AC}$ | [6] | ||
| $\psi_{\Omega_1}(\Omega^\Omega)=\Gamma_0$ | $\text{ATR}_0,\Delta^1_1-\text{CA}+\text{BR}$ | $\text{KPi}^-,\text{CZF}^-+\exists\kappa(\kappa\text{ is inaccessible})$ | [7] | |
| $\theta(\delta_n,0)$ | $\text{ACA}_0+(\Pi^1_{n+1}-\text{BI})$ | $\text{KP}^-+(\Pi_{n+1}-\text{Foundation})$ | [8] | $\delta_1=\Omega^\omega,\delta_{n+1}=\Omega^{\delta_n}$ |
| $\theta(\eta_n,0)$ | $\text{ACA}+(\Pi^1_{n+1}-\text{BI})$ | $\text{KP}^-+\text{IND}+(\Pi_{n+1}-\text{Foundation})$ | [9] | $\eta_1=\Omega^{\varepsilon_0},\eta_{n+1}=\Omega^{\eta_n}$ |
| $\psi_{\Omega_1}(\varepsilon_{\Omega+1})$ | $\text{ACA}+\text{BI}$ | $\text{KP}$ | [10] | Bachmann-Howard ordinal |
| $\psi_{\Omega_1}(\Omega_\omega)$ | $\Pi^1_1-\text{CA}_0, \Delta^1_2-\text{CA}_0$ | [11] | ||
| $\psi_{\Omega_1}(\Omega_\omega\varepsilon_0)$ | $\Pi^1_1-\text{CA}$ | [12] | ||
| $\psi_{\Omega_1}(\varepsilon_{\Omega_\omega+1})$ | $\Pi^1_1-\text{CA}+\text{BI}$ | $\text{KPl}$ | [13] | Takeuti-Feferman-Buchholz ordinal |
| $\psi_{\Omega_1}(\Omega_{\omega^\omega})$ | $\Delta^1_2-\text{CR}$ | [14] | ||
| $\psi_{\Omega_1}(\Omega_{\varepsilon_0})$ | $\Delta^1_2-\text{CA}$ | [15] | ||
| $\psi_{\Omega_1}(\varepsilon_{\mathcal{I}+1})$ | $\Delta^1_2-\text{CA}+\text{BI}$ | $\text{KPi}$ | [16] | $\mathcal{I}$ is the least weakly inaccessible cardinal |
| $\psi_{\Omega_1}(\Omega_{\mathcal{I}_\omega})$ | $\text{KPh}$ | |||
| $\psi_{\Omega_1}(\varepsilon_{\mathcal{M}+1})$ | $\Delta^1_2-\text{CA}+\text{BI}+\text{(M)}$ | $\text{KPM}$ | [17][18] | $\mathcal{M}$ is the least weakly Mahlo cardinal |
| $\psi_{\Omega_1}(\Omega_{\mathcal{M}+\omega})$ | $\text{KPM}^+$ | |||
| $\Psi^0_{\Omega_1}(\varepsilon_{\mathcal{K}+1})$ | $\text{ACA}+\text{BI}+(\Pi^1_4-\beta\text{-model Reflection})$ | $\text{KP}+(\Pi_3-\text{Reflection})$ | [19] | $\mathcal{K}$ is the least $\Pi^1_1$-indescribable cardinal |
| $\Psi^{\varepsilon_{\Xi+1}}_\mathbb{X}$ | $\text{ACA}+\text{BI}+\beta\text{-model Reflection}$ | $\text{KP}+(\Pi_\omega-\text{Reflection})$ | [20] | $\Xi$ is the least $\Pi^2_0$-indescribable cardinal |
| $\Psi^{\varepsilon_{\Upsilon+1}}_\mathbb{H}$ | $\text{Stability},\text{KPi}+\forall\alpha\exists\kappa$ $L_\kappa\prec_1 L_{\kappa+\alpha}$ | [21][22] | $\Upsilon$ is the least subtle cardinal | |
| $\Psi^{\varepsilon_{\mathbf{\text{I}}+1}}_\mathbb{K}$ | $\Delta^1_2-\text{CA}+\text{BI}+\text{ parameter-free}$ $\Pi^1_2-\text{CA}$ | $\text{KP}+\exists M(\text{Trans(M)}\land M\prec_1 V)$ | [23] | See (*) |
Notes
(*) $\mathbf{\text{I}}$ is not the least weakly inaccessible $\mathcal{I}$. It has a somewhat technical definition; cf the reference given.
