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List of ordinals: Difference between revisions
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* [[1]], the first successor ordinal
* [[omega|\( \omega \)]], the first limit ordinal
* [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal
* [[omega^3|\( \omega^{3} \)]]
* [[omega^omega|\( \omega^{\omega} \)]]
* [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA<sub>0</sub><sup>(sort out page)</sup>
* [[veblen_hierarchy#zeta|\( \psi_{0}(\Omega^{2}) = \varphi(2,0) = \zeta_{0} \)]]<sup>(decide if own page)</sup>
* [[veblen_hierarchy|\( \psi_{0}(\Omega^{\omega}) = \varphi(\omega,0) \)]], the second infinite primitive recursively principal ordinal
* [[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<sup>(decide if keep)</sup>
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* [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal)
* [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal)
* [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension
* [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction
* [[Bird's ordinal|\( \psi_{0}(\Omega_\Omega) \)]], sometimes known as Bird's ordinal
* [[extended_buchholz_ordinal|\( \psi_{0}(\Omega_{\Omega_{\dots}}) \)]], the EBO (Extended Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-transfinite recursion
* \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below)
* \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM
* \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF)
* \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L
* <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals
* PTO of \( \Pi^1_2 \)-comprehension
* PTO of \( \text{Z}_{2} \)
* PTO of \( \text{KP} + "\omega_1 \) exists \( " \)
* PTO of \( \text{ZFC} \)
* [[church_kleene_ordinal|\( \omega^{\text{CK}}_{1} \)]], the Church-Kleene ordinal, i.e. the least ordinal which is not recursive and the second admissible ordinal
* <nowiki>\( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least height of a model of \( \Pi^1_1 \)-comprehension</nowiki>
* The least recursively inaccessible ordinal = the least height of a model of \( \textsf{KPi} \) or \( \Delta^1_2 \)-comprehension<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>
* The least recursively Mahlo ordinall = the least doubly \( \Pi_2 \)-reflecting ordinal = the least height of a model of \( \textsf{KPM} \)<ref name=":0" /><sup>(p.3)</sup>
* 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>
* The least \( \Pi_n \)-reflecting ordinals, for \( 2<n<\omega \)<ref name=":1" />
* The least \( (+1) \)-stable ordinal<ref name=":0" /><sup>(p.4)</sup>
* The least \( (
* The least \( (\cdot 2) \)-stable ordinal
* The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" /><sup>(p.4)</sup>
* The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky-->
* The least \( \Sigma^1_1 \)-reflecting ordinal = the least non-Gandy ordinal<ref name="OrderOfReflection" /><sup>(pp.3,9)</sup><ref name=":0" />
* The \( (\sigma^1_1)^n \)-reflecting ordinals for \( 1<n<\omega \)<!--iterated \( \Sigma^1_1 \)-reflection--><ref name="OrderOfReflection" /><sup>(p.20)</sup>
* The least \( (^++1) \)-stable ordinal<sup>(Is this strictly greater than previous entry?)</sup><ref name="OrderOfReflection" /><sup>(p.20)</sup>
* The least doubly \( (+1) \)-stable ordinal<ref name=":0" /><sup>(p.4)</sup>
* The least \(\omega\)-ply stable ordinal = the least ordinal stable up to a nonprojectible ordinal
* The least
* The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" /><sup>(pp.5-6)</sup> = least ordinal \( \Pi_3 \)-reflecting on the ordinals stable up to it<ref name=":2" /><sup>(p.221)</sup>
* HIGHER STABILITY STUFF GOES HERE<sup>(sort out)</sup>
* Some Welch stuff here
* [[Infinite_time_Turing_machine|Infinite time Turing machine]] ordinals
** \( \lambda \), the supremum of all writable ordinals = the least ordinal stable up to \( \zeta \) ordinal
** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \)
** \( \zeta \), the supremum of all eventually writable ordinals = the least \( \Sigma_2 \)-extendible ordinal
** \( \Sigma \), the supremum of all accidentally writable ordinals = the least target of \( \Sigma_2 \)-stability
* Welch's \(E_0\)-ordinals <ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref>
* Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" />
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