List of ordinals: Difference between revisions
More ordinals
(Most common notation instead of xkcdforums notation) |
(More ordinals) |
||
Line 1:
==== Countable ordinals ====
In this list we assume there is a transitive model of ZFC.<!--Necessary to compare ordinals such as "least a such that L_a models ZFC"-->
* [[0]], the smallest ordinal
* [[1]], the first successor ordinal
Line 41 ⟶ 42:
* The least \( \Sigma_2 \)-admissible ordinal<ref name=":0" /><sup>(pp.5-6)</sup> = least ordinal \( \Pi_3 \)-reflecting on class of stable ordinals below<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
Line 46 ⟶ 48:
** \( \zeta \), the supremum of all eventually writable ordinals
** \( \Sigma \), the supremum of all accidentally writable ordinals
* The smallest [[gap_ordinals|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>
* Least start of a gap in the constructible universe of length 2<ref name="Gaps" />
* Least \( \beta \) that starts a gap of length \( \beta \)<ref name="Gaps" />
* Least \( \beta \) that starts gap of length \( \beta^\beta \)<ref name="Gaps" />
* 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 StackExcahnge 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>
* 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>
==== Uncountable ordinals ====
| |||