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

* \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π12 Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>p.24

* \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π12 Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref>p.24

* 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>(p.3)

* 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" />(p.3)

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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
-* \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension =  = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π<sup>1</sup><sub>2</sub> Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref><sup>p.24</sup>
+* \( \omega^{\text{CK}}_{\omega} \), the least limit of admissible ordinals = the least \(\alpha\) such that \(L_\alpha\cap\mathcal P(\omega)\) models \( \Pi^1_1 \)-comprehension = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_1\mathrm{-CA}_0\)<ref name="Pi12Consequences">J. P. Aguilera, F. Pakhomov, "[https://arxiv.org/abs/2109.11652v1 The Π<sup>1</sup><sub>2</sub> Consequences of a Theory]" (2021). Accessed 18 January 2024.</ref><sup>p.24</sup>
 * 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>

List of ordinals: Difference between revisions

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