List of ordinals: Difference between revisions
Jump to navigation
Jump to search
Content added Content deleted
Augigogigi (talk | contribs) mNo edit summary Tag: Manual revert |
Augigogigi (talk | contribs) No edit summary |
||
Line 4: | Line 4: | ||
* [[1]], the first successor ordinal |
* [[1]], the first successor ordinal |
||
* [[omega|\( \omega \)]], the first limit ordinal |
* [[omega|\( \omega \)]], the first limit ordinal |
||
* [[omega*2|\( \omega\cdot2 \)]] |
|||
* [[omega^2|\( \omega^{2} \)]] |
* [[omega^2|\( \omega^{2} \)]] |
||
* [[omega^3|\( \omega^{3} \)]] |
* [[omega^3|\( \omega^{3} \)]] |
||
* [[omega^omega|\( \omega^{\omega} \)]] |
* [[omega^omega|\( \omega^{\omega} \)]] |
||
* [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]] |
* [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]]<sup>(sort out page)</sup> |
||
* [[the_veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]] |
* [[the_veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]<sup>(decide if own page)</sup> |
||
* [[the_veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] |
* [[the_veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] |
||
* [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of |
* [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of ATR<sub>0</sub> |
||
* [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal |
* [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal<sup>(decide if keep)</sup> |
||
* [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi(1@\omega) \)]], the SVO (Small Veblen ordinal) |
* [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi(1@\omega) \)]], the SVO (Small Veblen ordinal) |
||
* [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) |
* [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) |
||
Line 19: | Line 18: | ||
* [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFB (Takeuti-Feferman-Buchholz ordinal) |
* [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFB (Takeuti-Feferman-Buchholz ordinal) |
||
* [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) |
* [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) |
||
* \( \psi_{\Omega}(\varepsilon_{I+1}) \), the PTO of |
* \( \psi_{\Omega}(\varepsilon_{I+1}) \), the PTO of KPi |
||
* \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of |
* \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM |
||
* \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of \( |
* \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. |
||
* \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of |
* \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L |
||
* \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals |
* \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals |
||
* PTO of \( \text{Z}_{2} \) |
* PTO of \( \text{Z}_{2} \) |
||
* PTO of \( \text{ZFC} \) |
* PTO of \( \text{ZFC} \) |
||
* [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] |
* [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] |
||
* RECURSIVE ORDINALS GO HERE |
* RECURSIVE ORDINALS GO HERE<sup>(sort out)</sup> |
||
* STABILITY STUFF GOES HERE |
* STABILITY STUFF GOES HERE<sup>(sort out)</sup> |
||
* [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals |
* [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals |
||
** \( \lambda \), the supremum of all writable ordinals |
** \( \lambda \), the supremum of all writable ordinals |
||
Line 35: | Line 34: | ||
** \( \Sigma \), the supremum of all accidentally writable ordinals |
** \( \Sigma \), the supremum of all accidentally writable ordinals |
||
* The smallest [[gap_ordinals|gap ordinal]] |
* The smallest [[gap_ordinals|gap ordinal]] |
||
* MORE STUFF GOES HERE<sup>(sort out)</sup> |
|||
==== Uncountables ==== |
==== Uncountables ==== |
Revision as of 22:09, 5 September 2022
Countables
- 0, the smallest ordinal
- 1, the first successor ordinal
- \( \omega \), the first limit ordinal
- \( \omega^{2} \)
- \( \omega^{3} \)
- \( \omega^{\omega} \)
- \( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)(sort out page)
- \( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)(decide if own page)
- \( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)
- \( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \), the Feferman-Schutte ordinal and the PTO of ATR0
- \( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \), the Ackermann Ordinal(decide if keep)
- \( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi(1@\omega) \), the SVO (Small Veblen ordinal)
- \( \psi_{0}(\Omega^{\Omega^{\Omega}}) \), the LVO (Large Veblen ordinal)
- \( \psi_{0}(\Omega_{2}) \), the BHO (Bachmann-Howard ordinal)
- \( \psi_{0}(\Omega_{\omega}) \), the BO (Buchholz ordinal)
- \( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \), the TFB (Takeuti-Feferman-Buchholz ordinal)
- \( \psi_{0}(\Omega_{\Omega_{\dots}}) \), the EBO (Extended Buchholz ordinal)
- \( \psi_{\Omega}(\varepsilon_{I+1}) \), the PTO of KPi
- \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM
- \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl.
- \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L
- \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second OCF using indescribable cardinals
- PTO of \( \text{Z}_{2} \)
- PTO of \( \text{ZFC} \)
- \( \omega^{\text{CK}}_{1} \)
- RECURSIVE ORDINALS GO HERE(sort out)
- STABILITY STUFF GOES HERE(sort out)
- Infinite time Turing machine ordinals
- \( \lambda \), the supremum of all writable ordinals
- \( \gamma \), the supremum of all clockable ordinals
- \( \zeta \), the supremum of all eventually writable ordinals
- \( \Sigma \), the supremum of all accidentally writable ordinals
- The smallest gap ordinal
- MORE STUFF GOES HERE(sort out)
Uncountables
- \( \Omega \), the smallest uncountable ordinal
- \( I \), the smallest inaccessible ordinal
- \( M \), the smallest mahlo cardinal
- \( K \), the smallest weakly compact cardinal