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{{stub}}An ordinal is called to be '''admissible''' if \(L_\alpha\) is an {{w|admissible set}}, where \(L\) is {{w|Gödel's constructible universe}}.<ref>[https://core.ac.uk/download/pdf/82692209.pdf Gostanian, Richard. The next admissible ordinal.]</ref> For successor ordinal \(\alpha\), \(1+\alpha\)-th admissible ordinal is denoted by \(\omega_\alpha^\text{CK}\). For example, \(\omega_\omega^\text{CK}\) is not admissible but it is a limit of admissibles.
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{{stub}}An ordinal is called to be '''admissible''' if \(L_\alpha\) is an {{w|admissible set}}, where \(L\) is [[Rank-into-rank_cardinal#Constructible_universe|Gödel's constructible universe]].<ref>[https://core.ac.uk/download/pdf/82692209.pdf Gostanian, Richard. The next admissible ordinal.]</ref> For successor ordinal \(\alpha\), \(1+\alpha\)-th admissible ordinal is denoted by \(\omega_\alpha^\text{CK}\). For example, \(\omega_\omega^\text{CK}\) is not admissible but it is a limit of admissibles.
 
== Sources ==
 
== Sources ==
 
<references/>
 
<references/>
 
== See also ==
 
== See also ==
 
{{Transfinite ordinals}}
 
{{Transfinite ordinals}}
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[[Category:Transfinite ordinals]]

Revision as of 08:38, January 16, 2020

An ordinal is called to be admissible if \(L_\alpha\) is an admissible set, where \(L\) is Gödel's constructible universe.[1] For successor ordinal \(\alpha\), \(1+\alpha\)-th admissible ordinal is denoted by \(\omega_\alpha^\text{CK}\). For example, \(\omega_\omega^\text{CK}\) is not admissible but it is a limit of admissibles.

Sources

  1. Gostanian, Richard. The next admissible ordinal.

See also

Ordinals, ordinal analysis and set theory

Basics: cardinal numbers · normal function · ordinal notation · ordinal numbers · fundamental sequence · ordinal collapsing function
Theories: Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\Gamma_0\) · \(\vartheta(\Omega^3)\) · \(\vartheta(\Omega^\omega)\) · \(\vartheta(\Omega^\Omega)\) · \(\vartheta(\varepsilon_{\Omega + 1}) = \psi(\Omega_2)\) · \(\psi(\Omega_\omega)\) · \(\psi(\varepsilon_{\Omega_\omega + 1})\) · \(\psi(\psi_I(0))\)‎ · \(\omega_1^\mathfrak{Ch}\) · \(\omega_1^\text{CK}\) · \(\omega_\alpha^\text{CK}\) · \(\lambda,\zeta,\Sigma,\gamma\) · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Slow-growing hierarchy · Hardy hierarchy · Middle-growing hierarchy · N-growing hierarchy
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal · more...

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