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
* <nowiki>\( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least
* The least recursively inaccessible ordinal = the least
* The least recursively Mahlo ordinal = the least doubly \( \Pi_2 \)-reflecting ordinal = the least
* 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" />
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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>
* Least start of third-order gap = least height of 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
* 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>
* 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 \(\omega_1\), however would still be greater than the least height of a model of KP+"\( \omega_1 \) exists".
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