List of ordinals: Difference between revisions
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* [[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>. This is the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) satisfies Feferman's theory \(\mathrm{IR}\).<ref>S. G. Simpson, "[https://sgslogic.net/t20/talks/feferfest/paper3.pdf Predicativity: The Outer Limits]" (2000), p.3. Accessed 30 January 2024.</ref>
* [[the_veblen_hierarchy#ackermann|\( \psi_{0}(\Omega^{\Omega^{2}}) = \varphi(1,0,0,0) \)]], the Ackermann Ordinal<sup>(decide if keep)</sup>
* [[small_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\omega}}) = \varphi\begin{pmatrix}1 \\ \omega\end{pmatrix} \)]], the SVO (Small Veblen ordinal)
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* 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
* The least (next recursively inaccessible ordinal)-stable ordinal
* The least (next recursively Mahlo ordinal)-stable ordinal
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** \( \zeta \), 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
* The least ordinal in \(
* The least ordinal in \(E_\eta\), for \(\eta > 1\)<ref name="Welch2010Draft" />
* The least admissible \(\alpha\) so that \(L_\alpha\) satisfies
* Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" />
* The smallest [[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> = the least simultaneously \(\Sigma_n\)-nonprojectible ordinal for all \(n<\omega\)<ref>The Higher Infinite in Proof Theory, Michael Rathjen</ref>
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* Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" />
* Least \( \beta \) that starts gap of length \( \beta^+ \) - \(L_{\beta^+}\) here is a 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>
<!-- How large is the ordinal theta on page 74 of Hachtman's "Calibrating Determinacy Strength in Borel Hierarchies" (https://escholarship.org/content/qt6tk9351b/qt6tk9351b_noSplash_82b2a392eeaa5314b2f174d8d2ae832b.pdf)?-->
* Least start of third-order gap = least \(\beta\) such that \(L_\beta\) is a model of \( ZFC^- \)+"\( \beth_1 \) exists"<ref name=":0" /><sup>(p.6)</sup>
* <!--I would like to put "least start of fourth-order gap" here but my only source is a StackExchange answer-->
* Least height of model of ZFC<ref name=":0" /><sup>(p.6)</sup>
<!--* Least height of a β-model of \(\mathsf{GBC}+\Pi^1_2\mathsf{-CA}\)<ref>K. J. Williams, ''[https://arxiv.org/abs/1804.09526 The Structure of Models of Second-order Set Theories]'' (pp.107--108). PhD dissertation, 2018.</ref><ref>This ordinal is larger than the previous, as the first-order part of any model of \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) contains a model of ZFC in its first-order part (see Williams18 p.8) and this is larger than the ordinal for \(\mathsf{GBC}+\Pi^1_1\mathsf{-CA}\) (see p.108). It is smaller than the next ordinal, as Williams18 axiomatizes GBC using first-order logic, and the least stable ordinal is at least \(\mathrm{sup}\{\mu\alpha.L_\alpha\vDash T\mid T\textrm{ is a recursive first-order set theory}\}\).</ref>-->
* 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 β₂-models of full second order arithmetic and some related facts] (1974, Fundamenta Mathematicae 82(2), pp.175-189). Accessed 7 September 2022.</ref>, this equals the supremum of the \(\Sigma^1_2\)-soundness ordinals of recursively enumerable \(\Sigma^1_2\)-sound extensions of \(\mathrm{ACA}_0\)<ref name="Pi12Consequences" /><sup>p.23</sup>
* Least stable ordinal that's also during a gap - height of least \( \beta_2 \)-model of \( Z_2 \)<ref name="beta2Models" />
* 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>
The least ordinal \(\alpha\) so that \(\alpha\) is uncountable in \(L\) is equal to the least ordinal which starts a gap of length \(\omega_1\). If \(V = L\), then this is greater than all ordinals on this list and equal to \(\Omega\), while if \(0^\sharp\) exists it is ''significantly'' smaller than \(\omega_1\) and also smaller than the least \(\Pi^1_3\)-reflecting and \(\Sigma^1_3\)-reflecting ordinals<ref>J. P. Aguilera, C. B. Switzer, "[https://arxiv.org/abs/2311.12533v1 Reflection Properties of Ordinals in Generic Extensions]", p.18</ref>, however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists". Also assuming projective determinacy, for \(\alpha<\omega_1^{M_n}\), \(\alpha\) is \(M_n\)-stable iff it is \(\Sigma^1_{n+2}\)-reflecting when \(n\) is even, and \(\Pi^1_{n+2}\)-reflecting when \(n\) is odd, \(M_n\) as in the extender model. <ref>J. P. Aguilera, "[https://www.dropbox.com/s/lrdm0wxscry7ehj/RLPO.pdf?dl=0 Recursively Large Projective Ordinals]", 2022. Accessed 19 January 2024.</ref><sup>Corollary 21</sup> <!--How large are the ordinals \(\sigma_{\Sigma_n,\omega_2}\) and \(\sigma_{\Pi_n,\omega_2}\) here, assuming 0# exists? https://arxiv.org/pdf/2311.12533v1.pdf#page=19-->
== Uncountable ordinals ==
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