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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.
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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.
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It is believed that \(f_{\omega_\alpha^\text{CK}}(n)\) is approximated to \(\Sigma_\alpha(n)\), the \(\alpha\)-th order [[busy beaver function]] using some unspecified system of [[fundamental sequence]]s. Due to many ways of defining fundamental sequences and busy beaver functions with oracles it may be not true.
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== Sources ==
 
== Sources ==
 
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<references/>

Revision as of 09:00, 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.

It is believed that \(f_{\omega_\alpha^\text{CK}}(n)\) is approximated to \(\Sigma_\alpha(n)\), the \(\alpha\)-th order busy beaver function using some unspecified system of fundamental sequences. Due to many ways of defining fundamental sequences and busy beaver functions with oracles it may be not true.

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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