List of ordinals: Difference between revisions
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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 ATR<sub>0</sub> |
* [[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> |
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* [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal<sup>(decide if keep)</sup> |
* [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal<sup>(decide if keep)</sup> |
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* [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi |
* [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal) |
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* [[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) |
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* [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) |
* [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal) |
Revision as of 02:12, 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)
- \( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \), 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 \( (^+) \)-stable ordinal = least \( \Pi^1_1 \)-reflecting ordinal[1](p.4)
- The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals[3]
- The least \( \Sigma^1_1 \)-reflecting ordinal = least non-Gandy ordinal[3](pp.3,9)[1]
- The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)[3](p.20)
- The least \( (^++1) \)-stable ordinal(Is this strictly greater than previous entry?)[3](p.20)
- The least doubly \( (+1) \)-stable ordinal[1](p.4)
- The least nonprojectible ordinal[1](p.5) = least ordinal \( \Pi_2 \)-reflecting on class of stable ordinals below[4](p.218)
- The least \( \Sigma_2 \)-admissible ordinal[1](pp.5-6) = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below[4](p.221)
- 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 1.6 1.7 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.
- ↑ 3.0 3.1 3.2 3.3 J. P. Aguilera, The Order of Reflection (2019, arxiv preprint). Accessed 7 September 2022.
- ↑ 4.0 4.1 E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225 Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.