List of ordinals: Difference between revisions
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#REDIRECT [[List of countable ordinals]] |
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==== Countables ==== |
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* [[0]], the smallest ordinal |
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* [[1]], the first successor ordinal |
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* [[omega|\( \omega \)]], the first limit ordinal |
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* [[omega*2|\( \omega\cdot2 \)]] |
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* [[omega^2|\( \omega^{2} \)]] |
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* [[omega^3|\( \omega^{3} \)]] |
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* [[omega^omega|\( \omega^{\omega} \)]] |
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* [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]] |
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* [[the_veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]] |
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* [[the_veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]] |
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* [[feferman-schutte_ordinal|\( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \)]], the Feferman-Schutte ordinal and the PTO of \( \text{ATR}_{0} \) |
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* [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal |
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* [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi(1@\omega) \)]], the SVO (Small Veblen ordinal) |
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* [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal) |
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* [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) |
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* [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) |
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* [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFB (Takeuti-Feferman-Buchholz ordinal) |
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* [[extened_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) |
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* \( \psi_{\Omega}(\varepsilon_{I+1}) \), the PTO of \( \text{KPi} \) |
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* \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of \( \text{KPM} \) |
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* \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of \( \text{KP} + \Pi_{3}\text{-refl.} \) |
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* \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of \( \text{KP} \) with a \( \Pi_{\mathbb{N}}\text{-refl.} \) universe under ZF + V = L |
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* \( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \), the limit of Jan-Carl Stegert's second [[ordinal_collapsing_function|OCF]] using indescribable cardinals |
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* PTO of \( \text{Z}_{2} \) |
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* PTO of \( \text{ZFC} \) |
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* [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] |
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* RECURSIVE ORDINALS GO HERE |
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* STABILITY STUFF GOES HERE |
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* [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals |
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** \( \lambda \), the supremum of all writable ordinals |
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** \( \gamma \), the supremum of all clockable ordinals |
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** \( \zeta \), the supremum of all eventually writable ordinals |
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** \( \Sigma \), the supremum of all accidentally writable ordinals |
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* The smallest [[gap_ordinals|gap ordinal]] |
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==== Uncountables ==== |
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* [[omega_1|\( \Omega_{1} \), the smallest [[uncountable ordinal]] |
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Revision as of 21:48, 5 September 2022
Countables
- 0, the smallest ordinal
- 1, the first successor ordinal
- \( \omega \), the first limit ordinal
- \( \omega\cdot2 \)
- \( \omega^{2} \)
- \( \omega^{3} \)
- \( \omega^{\omega} \)
- \( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)
- \( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)
- \( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)
- \( \psi_{0}(\Omega^{\Omega}) = \varphi(1,0,0) = \Gamma_{0} \), the Feferman-Schutte ordinal and the PTO of \( \text{ATR}_{0} \)
- \( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \), the Ackermann Ordinal
- \( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi(1@\omega) \), 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 \( \text{KPi} \)
- \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of \( \text{KPM} \)
- \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of \( \text{KP} + \Pi_{3}\text{-refl.} \)
- \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of \( \text{KP} \) with a \( \Pi_{\mathbb{N}}\text{-refl.} \) 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
- STABILITY STUFF GOES HERE
- 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
Uncountables
- [[omega_1|\( \Omega_{1} \), the smallest uncountable ordinal