List of ordinals: Difference between revisions
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==== Countable ordinals ==== |
==== Countable ordinals ==== |
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In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"--> |
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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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* The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" /><sup>(pp.5-6)</sup> = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below<ref name=":2" /><sup>(p.221)</sup> |
* The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" /><sup>(pp.5-6)</sup> = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below<ref name=":2" /><sup>(p.221)</sup> |
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* HIGHER STABILITY STUFF GOES HERE<sup>(sort out)</sup> |
* HIGHER STABILITY STUFF GOES HERE<sup>(sort out)</sup> |
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* Some Welch stuff here |
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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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** \( \zeta \), the supremum of all eventually writable ordinals |
** \( \zeta \), the supremum of all eventually writable ordinals |
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** \( \Sigma \), the supremum of all accidentally writable ordinals |
** \( \Sigma \), the supremum of all accidentally writable ordinals |
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* The smallest [[gap_ordinals|gap ordinal]]<ref name="Gaps">W. Marek, M. Srebrny, [https://www.sciencedirect.com/science/article/pii/0003484374900059 Gaps in the Constructible Universe] (1973). Accessed 7 September 2022.</ref> |
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* The smallest [[gap_ordinals|gap ordinal]] |
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* Least start of a gap in the constructible universe of length 2<ref name="Gaps" /> |
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* MORE STUFF GOES HERE<sup>(sort out)</sup> |
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* Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" /> |
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* Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" /> |
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* Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".<ref name=":0" /><sup>(p.6)</sup> <ref>T. Arai, [https://arxiv.org/abs/1102.0596 A Sneak Preview of Proof Theory of Ordinals] (1997, preprint, p.17). Accessed 7 September 2022.</ref> |
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* Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" /><sup>(p.6)</sup> |
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* <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExcahnge answer--> |
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* Least height of model of ZFC<ref name=":0" /><sup>(p.6)</sup> |
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* Least stable ordinal<ref name=":0" /><sup>(p.6)</sup><ref name="SpectrumOfL">W. Marek, K. Rasmussen, [http://pldml.icm.edu.pl/pldml/element/bwmeta1.element.zamlynska-1afccedc-bd3e-45b8-a2f9-3cbb4c6000bb/c/rm21101.pdf Spectrum of L] </ref><sup>(p.9)</sup>, this is a limit of gap ordinals<ref name="beta2Models">W. Marek, [http://matwbn.icm.edu.pl/ksiazki/fm/fm82/fm82112.pdf Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref> |
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* Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models /> |
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* The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).<ref name="SpectrumOfL" /><sup>(p.8)</sup> |
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==== Uncountable ordinals ==== |
==== Uncountable ordinals ==== |
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Revision as of 02:50, 8 September 2022
Countable ordinals
In this list we assume there is a transitive model of ZFC.
- 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)
- Some Welch stuff 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[5]
- Least start of a gap in the constructible universe of length 2[5]
- Least \( \beta \) that starts a gap of length \( \beta \)[5]
- Least \( \beta \) that starts gap of length \( \beta^\beta \)[5]
- Least \( \beta \) that starts gap of length \( \beta^+ \) = least height of model of KP+"\( \omega_1 \) exists".[1](p.6) [6]
- Least start of third-order gap = least height of model of \( ZFC^- \)+"\( \beth_1 \) exists"[1](p.6)
- Least height of model of ZFC[1](p.6)
- Least stable ordinal[1](p.6)[7](p.9), this is a limit of gap ordinals[8]
- Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)[8]
- The least non-analytical ordinal. This is the least \( \alpha \) such that \( L_\alpha \prec L_{\omega_1} \).[7](p.8)
Uncountable ordinals
- \( \Omega \), the smallest uncountable ordinal
- \( I \), the smallest inaccessible ordinal
- \( M \), the smallest mahlo cardinal
- \( K \), the smallest weakly compact cardinal
- ↑ 1.00 1.01 1.02 1.03 1.04 1.05 1.06 1.07 1.08 1.09 1.10 1.11 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.
- ↑ 5.0 5.1 5.2 5.3 W. Marek, M. Srebrny, Gaps in the Constructible Universe (1973). Accessed 7 September 2022.
- ↑ T. Arai, A Sneak Preview of Proof Theory of Ordinals (1997, preprint, p.17). Accessed 7 September 2022.
- ↑ 7.0 7.1 W. Marek, K. Rasmussen, Spectrum of L
- ↑ 8.0 8.1 W. Marek, Stable sets, a characterization of \( \beta_2 \)-models of full second order arithmetic and some related facts (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.