List of ordinals: Difference between revisions
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Apeirology primarily studies the structure of ordinals. This study can be crudely split into three parts: the [[Recursive ordinal|recursive ordinals]] have an explicit (recursive) wellordering describing them; the nonrecursive, countable ordinals, where phenomena such as admissibility and reflection starts to arise; and the uncountable cardinals, particularly large cardinals, where many similarities to the [[natural numbers]] disappear and the primary objects being studied include elementary embeddings, [[cofinality]], [[Cardinal|cardinality]] and abstract reflection or partition properties.
As such, apeirology is linked to:
* [[Set theory]] (which includes study of large cardinals)
* [[A-recursion theory|α-recursion theory]] (the study of generalising recursion on the natural numbers to on L_α for [[admissible]] ordinals α)
* [[B-recursion theory|β-recursion theory]] (the generalisation of α-recursion theory to non-admissible α)
* [[Proof theory]] and [[ordinal analysis]] (which assigns recursive ordinals to theories according to the lengths of the recursive well-orders they can prove well-founded)
* Googology (which translates recursive ordinal notations into systems for constructing large finite numbers).
Below we list some milestone ordinals.
== Countable ordinals ==
In this list we assume there is a transitive model of ZFC
* [[0]], the smallest ordinal
* [[1]], the first successor ordinal
* [[omega|\( \omega \)]], the first limit ordinal
* [[omega^2|\( \omega^{2} \)]], the second infinite additive principal ordinal
*
*<!--w^3 is not significant enough to have its own page, other than being the PTO of EFA and ID0 + Exp.-->
* [[omega^omega|\( \omega^{\omega} \)]]
* [[epsilon_numbers#zero|\( \psi_{0}(\Omega) = \varphi(1,0) = \varepsilon_{0} \)]], the PTO of PA and ACA<sub>0</sub><sup>(sort out page)</sup>
* [[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)
* [[large_veblen_ordinal|\( \psi_{0}(\Omega^{\Omega^{\Omega}}) \)]], the LVO (Large Veblen ordinal)
* [[bachmann_howard_ordinal|\( \psi_{0}(\Omega_{2}) \)]], the BHO (Bachmann-Howard ordinal)
* [[buchholz_ordinal|\( \psi_{0}(\Omega_{\omega}) \)]], the BO (Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension
* [[takeuti-feferman-buchholz_ordinal|\( \psi_{0}(\varepsilon_{\Omega_{\omega} + 1}) \)]], the TFBO (Takeuti-Feferman-Buchholz ordinal) and the PTO of \( \Pi^1_1 \)-comprehension with bar induction
* [[
* [[extended_buchholz_ordinal|\( \psi_{
* \( \psi_{\Omega}(\varepsilon_{\chi_1(0)+1}) \), the PTO of KPi or \( \Delta^1_2 \)-comprehension with transfinite induction (in Rathjen's Mahlo OCF, also applies to the one below)
* \( \psi_{\Omega}(\psi_{\chi_{\varepsilon_{M+1}}(0)}(0)) \), the PTO of KPM
* \( \Psi^{0}_{\Omega}(\varepsilon_{K+1}) \), the PTO of KP+\(\Pi_{3}\)-refl. (in Rathjen's weakly compact OCF)
* \( \psi_\Omega(\varepsilon_{\mathbb{K}+1}) \), the PTO of KP with a \( \Pi_{\mathbb{N}}\)-refl. universe under ZF + V = L
* <nowiki>\( \Psi_{\mathbb{X}}^{\varepsilon_{\Upsilon+1}} \) where \( \mathbb{X} = (\omega^+; \textsf{P}_0; \epsilon; \epsilon; 0) \), the limit of Jan-Carl Stegert's second </nowiki>[[ordinal_collapsing_function|OCF]] using indescribable cardinals
* PTO of \( \
* PTO of \( \text{Z}_{2} \) = PTO of \(\mathrm{ZFC}\) minus powerset
* PTO of \( \text{KP} + "\omega_1 \) exists \( " \)
* 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>
* 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>
* 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" />
* The least \( (+1) \)-stable ordinal<ref name=":0" /><sup>(p.4)</sup>
* The least \( (
* The least \( (\cdot 2) \)-stable ordinal
* The least \( (^+) \)-stable ordinal = the least \( \Pi^1_1 \)-reflecting ordinal<ref name=":0" /><sup>(p.4)</sup>
* The least ordinal that is \( \Pi^1_1 \)-reflecting on the \( \Pi^1_1 \)-reflecting ordinals<ref name="OrderOfReflection">J. P. Aguilera, [https://arxiv.org/pdf/1906.11769.pdf The Order of Reflection] (2019, arxiv preprint). Accessed 7 September 2022.</ref><!--An important ordinal to consider according to Taranovsky-->
* 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
* 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 = \(\Sigma^1_2\)-soundness ordinal of \(\Pi^1_2\mathrm{-CA}_0\)<ref name="Pi12Consequences" /><sup>p.24</sup>
* The least
* 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>
* 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 = the least ordinal stable up to \( \zeta \) ordinal
** \( \gamma \), the supremum of all clockable ordinals, equal to \( \lambda \)
** \( \zeta \), 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 \(E_1\),<ref name="Welch2010Draft">P. D. Welch, [https://web.archive.org/web/20130108001818/https://maths.bris.ac.uk/~mapdw/det17.pdf Weak Systems of Determinacy and Arithmetical Quasi-Inductive Definitions] (April 2010 draft). Accessed 11 January 2023.</ref> in unpublished work Welch has shown this is an ordinal referred to as \(\zeta^{\varnothing^{\blacktriangledown}}\)<ref>R. S. Lubarsky, "ITTMs with Feedback", in ''[http://wwwmath.uni-muenster.de/logik/Personen/rds/festschrift.pdf Ways of Proof Theory]'', edited by R. Schindler, Ontos Series in Mathematical Logic (2010, p.338).</ref>
* 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>▼
* The least ordinal in \(E_\eta\), for \(\eta > 1\)<ref name="Welch2010Draft" />
* The least admissible \(\alpha\) so that \(L_\alpha\) satisfies \(\mathsf{AQI}\), arithmetical quasi-induction = the least admissible \(\alpha\) so that, for all \(x \in \mathcal{P}(\omega) \cap L_\alpha\), there are \(\xi, \sigma < \alpha\) so that \(L_\xi[x] \prec_{\Sigma_2} L_\sigma[x]\)<ref name="Welch2010Draft" />{{verification failed}}
* Least \(\beta\) where \(L_\beta\) starts a chain of \(\Sigma_3\)-elementary substructures <ref name="Welch2010Draft" />
▲* The smallest [[
*<!-- I'd like to put "least a so that some ordinal is undefinable in L_a", which is the same as the "least a so that L_b < L_a for some b < a", which is obviously bigger than the least gap ordinal. I don't know exactly where it lies, though. -->
* 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 \( \beta^+ \)
<!-- 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
* <!--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 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
* 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 ==
* [[omega_1|\( \Omega \)]], the smallest [[
Further on, there lie [[Large cardinal|large cardinals]], so big that their existence is unprovable in ZFC (assuming its consistency), but which are useful if they do exist:
* \(
* \(
* \( K \), the smallest [[weakly compact cardinal]]
== References ==
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