List of ordinals: Difference between revisions
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(Some ordinals) |
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==== Countable ordinals ==== |
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* [[0]], the smallest ordinal |
* [[0]], the smallest ordinal |
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* [[1]], the first successor ordinal |
* [[1]], the first successor ordinal |
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* [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] |
* [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]] |
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* RECURSIVE ORDINALS GO HERE<sup>(sort out)</sup> |
* RECURSIVE ORDINALS GO HERE<sup>(sort out)</sup> |
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* The least recursively inaccessible ordinal<ref name=":0">D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals] (2017). Accessed 7 September 2022.</ref><sup>(p.3)</sup> |
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* The least recursively Mahlo ordinal<ref name=":0" /><sup>(p.3)</sup> |
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* The least recursively hyper-Mahlo ordinal<ref name=":1">W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063<nowiki> Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.</nowiki></ref><sup>(p.13)</sup> |
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* The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" /> |
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* The least \( (+1) \)-stable ordinal<ref name=":0" /><sup>(p.4)</sup> |
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* The least recursively |
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* The least \( (^+) \)-stable ordinal<ref name=":0" /><sup>(p.4)</sup> |
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* The least doubly \( (+1) \)-stable ordinal<ref name=":0" /><sup>(p.4)</sup> |
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* The least nonprojectible ordinal<ref name=":0" /><sup>(p.5)</sup> |
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* [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals |
* [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals |
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** \( \lambda \), the supremum of all writable ordinals |
** \( \lambda \), the supremum of all writable ordinals |
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* MORE STUFF GOES HERE<sup>(sort out)</sup> |
* MORE STUFF GOES HERE<sup>(sort out)</sup> |
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==== Uncountable ordinals ==== |
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* [[omega_1|\( \Omega \)]], the smallest [[uncountable_ordinal|uncountable ordinal]] |
* [[omega_1|\( \Omega \)]], the smallest [[uncountable_ordinal|uncountable ordinal]] |
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* \( I \), the smallest [[inaccessible_ordinal|inaccessible ordinal]] |
* \( I \), the smallest [[inaccessible_ordinal|inaccessible ordinal]] |
Revision as of 01:47, 8 September 2022
Countable ordinals
- 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)
- The least recursively inaccessible ordinal[1](p.3)
- The least recursively Mahlo ordinal[1](p.3)
- The least recursively hyper-Mahlo ordinal[2](p.13)
- The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)[2]
- The least \( (+1) \)-stable ordinal[1](p.4)
- The least recursively
- The least \( (^+) \)-stable ordinal[1](p.4)
- The least doubly \( (+1) \)-stable ordinal[1](p.4)
- The least nonprojectible ordinal[1](p.5)
- HIGHER 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)
Uncountable ordinals
- \( \Omega \), the smallest uncountable ordinal
- \( I \), the smallest inaccessible ordinal
- \( M \), the smallest mahlo cardinal
- \( K \), the smallest weakly compact cardinal
- ↑ 1.0 1.1 1.2 1.3 1.4 1.5 D. Madore, A Zoo of Ordinals (2017). Accessed 7 September 2022.
- ↑ 2.0 2.1 W. Richter, P. Aczel, [https://www.duo.uio.no/handle/10852/44063 Inductive Definitions and Reflecting Properties of Admissible Ordinals] (1973, preprint, Universitetet i Oslo). Accessed 7 September 2022.