List of ordinals: Difference between revisions
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* 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
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* The least recursively inaccessible ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPi} \) or \(L_\alpha\cap\mathcal P(\omega)\) models \( \Delta^1_2 \)-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 ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least \(\alpha\) such that \(L_\alpha\) models \( \textsf{KPM} \)<ref name=":0" /><sup>(p.3)</sup>
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* The least (next \( \Pi_n \)-reflecting ordinal)-stable ordinal, for \( 2<n<\omega \)
* 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 = \(\Pi^1_2\)-ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" /><sup>p.24</sup>
* The least nonprojectible ordinal<ref name=":0" /><sup>(p.5)</sup> = the least ordinal \( \Pi_2 \)-reflecting on the ordinals stable up to it = the least limit of \(\omega\)-ply stable ordinals<ref name=":2">E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225<nowiki> Reflection and Partition Properties of Admissible Ordinals] (1980). Accessed 7 September 2022.</nowiki></ref><sup>(p.218)</sup>
* 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>
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* <!--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 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>, this equals the supremum of the \(\Sigma^1_2\)-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>
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