Changes: Admissible ordinal

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An ordinal is called admissible if \(L_\alpha\) is an admissible set, where \((L_\xi)_{\xi\in\textrm{Ord}}\) is the constructible hierarchy.^[1]

Here, a set \(M\) is said to be admissible (as a structure of set theory) if \((M,\in)\) is a model of \(\textsf{KP}\). Readers should be careful that some authors use \(\textsf{KP}\omega\) instead of \(\textsf{KP}\),^[2] and the admissibility for \(\textsf{KP}\omega\) is not equivalent to that for \(\textsf{KP}\). For example, the first admissible ordinal with respect to the admissibility for \(\textsf{KP}\) is \(\omega\), while the first admissible ordinal with respect to the admissibility for \(\textsf{KP} \omega\) is \(\omega_1^{\textrm{CK}}\) (also the second admissible ordinal with respect to the admissibility for \(\textsf{KP}\)). In this article, we deal with the convention of the admissibility for \(\textsf{KP}\).

The enumeration function of the class of admissible ordinals and limits of admissible ordinals is denoted by \(\alpha \mapsto \omega_{\alpha}^{\textrm{CK}}\)^{[citation needed]}. For a successor ordinal \(\alpha\), \(\omega_\alpha^\text{CK}\) is the \(1+\alpha\)-th admissible ordinal. However, \(\omega_\omega^\text{CK}\) and other ordinals of the form \(\omega_\alpha^\text{CK}\) with limit \(\alpha\) are not necessarily admissible although they are limits of admissibles. In the case that \(\omega_\alpha^\textrm{CK}\) is both admissible and a limit of admissibles, it's also called a recursively inaccessible ordinal.^[3]

It was believed in this community without proofs 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. However, there are many ways of defining fundamental sequences (which are necessary to define the FGH), and busy beaver functions with oracles. In fact, a system of fundamental sequences such that the claim is false is known,^[4] while a system of fundamental sequences such as the claim is true is not known.

Properties

If \(\alpha\) is admissible, then there is no \(\Sigma_1\) projection with parameters in \(L_\alpha\) that maps some \(\gamma<\alpha\) to \(\alpha\)^[5]
If \(\alpha\) is admissible, then \(L_\alpha\) satisfies a certain \(\Delta_1\)-separation schema^[6].

Extension

An extension of the concept of admissibility is \(\Sigma_n\)-admissibility for natural \(n\). An ordinal \(\alpha\) is \(\Sigma_n\)-admissible if \(L_\alpha\) is rudimentary closed and satisfies the \(\Sigma_n\)-collection schema^[7]. Note that ordinals which are \(\Sigma_n\)-admissible need not satisfy \(\Sigma_n\)-separation, for example the least \(\Sigma_2\)-admissible ordinal doesn't satisfy \(\Sigma_2\)-separation^[8].

Sources

↑ Gostanian, Richard. The next admissible ordinal.
↑ Christoph Duchhardt, Thinning Operators and \(\Pi_4\)-Reflection, Munster university doctral thesis, 2008. Accessed 2021-04-16.
↑ D. Madore, A Zoo of Ordinals #2.3 (2017, accessed 2020-11-18)
↑ T. Kihara, omega-1-ck.pdf.
↑ R. Jensen, The fine structure of the constructible hierarchy (1972, accessed 2020-11-19) (p.254)
↑ K. Devlin, An introduction to the fine structure of the constructible hierarchy (1974) (p.19)
↑ E. Kranakis, Reflection and partition properties of admissible ordinals (p.216). Accessed 2021-03-27
↑ M. Rathjen, The Higher Infinite in Proof Theory (p.19). Accessed 2021-03-27

Basics: cardinal numbers · ordinal numbers · limit ordinals · fundamental sequence · normal form · transfinite induction · ordinal notation · Absolute infinity
Theories: Robinson arithmetic · Presburger arithmetic · Peano arithmetic · KP · second-order arithmetic · ZFC
Model Theoretic Concepts: structure · elementary embedding
Countable ordinals: \(\omega\) · \(\varepsilon_0\) · \(\zeta_0\) · \(\eta_0\) · \(\Gamma_0\) (Feferman–Schütte ordinal) · \(\varphi(1,0,0,0)\) (Ackermann ordinal) · \(\psi_0(\Omega^{\Omega^\omega})\) (small Veblen ordinal) · \(\psi_0(\Omega^{\Omega^\Omega})\) (large Veblen ordinal) · \(\psi_0(\varepsilon_{\Omega + 1}) = \psi_0(\Omega_2)\) (Bachmann-Howard ordinal) · \(\psi_0(\Omega_\omega)\) with respect to Buchholz's ψ · \(\psi_0(\varepsilon_{\Omega_\omega + 1})\) (Takeuti-Feferman-Buchholz ordinal) · \(\psi_0(\Omega_{\Omega_{\cdot_{\cdot_{\cdot}}}})\) (countable limit of Extended Buchholz's function)‎ · \(\omega_1^\mathfrak{Ch}\) (Omega one of chess) · \(\omega_1^{\text{CK}}\) (Church-Kleene ordinal) · \(\omega_\alpha^\text{CK}\) (admissible ordinal) · recursively inaccessible ordinal · recursively Mahlo ordinal · reflecting ordinal · stable ordinal · \(\lambda,\gamma,\zeta,\Sigma\) (Infinite time Turing machine ordinals) · gap ordinal · List of countable ordinals
Ordinal hierarchies: Fast-growing hierarchy · Hardy hierarchy · Slow-growing hierarchy · Middle-growing hierarchy · N-growing hierarchy
Ordinal functions: enumeration · normal function · derivative · Veblen function · ordinal collapsing function · Weak Buchholz's function · Bachmann's function · Madore's function · Feferman's theta function · Buchholz's function · Extended Weak Buchholz's function · Extended Buchholz's function · Jäger-Buchholz function · Jäger's function · Rathjen's psi function · Rathjen's Psi function · Stegert's Psi function · Arai's psi function
Uncountable cardinals: \(\omega_1\) · omega fixed point · inaccessible cardinal \(I\) · Mahlo cardinal \(M\) · weakly compact cardinal \(K\) · indescribable cardinal · rank-into-rank cardinal
Classes: \(\textrm{Card}\) · \(\textrm{On}\) · \(V\) · \(L\) · \(\textrm{Lim}\) · \(\textrm{AP}\) · Class (set theory)

[1] Gostanian, Richard. The next admissible ordinal.

[2] Christoph Duchhardt, Thinning Operators and \(\Pi_4\)-Reflection, Munster university doctral thesis, 2008. Accessed 2021-04-16.

[3] D. Madore, A Zoo of Ordinals #2.3 (2017, accessed 2020-11-18)

[4] T. Kihara, omega-1-ck.pdf.

[5] R. Jensen, The fine structure of the constructible hierarchy (1972, accessed 2020-11-19) (p.254)

[6] K. Devlin, An introduction to the fine structure of the constructible hierarchy (1974) (p.19)

[7] E. Kranakis, Reflection and partition properties of admissible ordinals (p.216). Accessed 2021-03-27

[8] M. Rathjen, The Higher Infinite in Proof Theory (p.19). Accessed 2021-03-27

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-An ordinal is called to be '''admissible''' if \(L_\alpha\) is an admissible set, where \((L_\xi)_{\xi\in\textrm{Ord}}\) is the [[constructible universe|constructible hierarchy]].<ref>{{cite web|first=Richard|last=Gostanian|url=https://core.ac.uk/download/pdf/82692209.pdf|title=The next admissible ordinal}}</ref>
+An [[ordinal]] is called '''admissible''' if \(L_\alpha\) is an admissible set, where \((L_\xi)_{\xi\in\textrm{Ord}}\) is the [[constructible universe|constructible hierarchy]].<ref>{{cite web|first=Richard|last=Gostanian|url=https://core.ac.uk/download/pdf/82692209.pdf|title=The next admissible ordinal}}</ref>
-Here, a set \(M\) is said to be '''admissible''' (as a structure of set theory) if \((M,\in)\) is a model of [[Kripke-Platek set theory|\(\textsf{KP}\)]]. Readers should be careful that some authors use \(\textsf{KP}\omega\) instead of \(\textsf{KP}\),<ref>Christoph Duchhardt, [https://repositorium.uni-muenster.de/document/miami/265b8283-850a-44a8-845d-f902ff1ca0fa/diss_duchhardt.pdf Thinning Operators and \(\Pi_4\)-Reflection], Munster university doctral thesis, 2008.</ref> and the admissibility for \(\textsf{KP}\omega\) is not equivalent to that for \(\textsf{KP}\). For example, the first admissible ordinal with respect to the admisibility for \(\textsf{KP}\) is \(\omega\), while the first admissible ordinal with respect to the admisibility for \(\textsf{KP} \omega\) is [[Church-Kleene ordinal|\(\omega_1^{\textrm{CK}}\)]] (also the second admissible ordinal with respect to the admissibility for \(\textsf{KP}\)). In this article, we deal with the convention of the admissibility for \(\textsf{KP}\).
+Here, a set \(M\) is said to be '''admissible''' (as a structure of set theory) if \((M,\in)\) is a model of [[Kripke-Platek set theory|\(\textsf{KP}\)]]. Readers should be careful that some authors use \(\textsf{KP}\omega\) instead of \(\textsf{KP}\),<ref>Christoph Duchhardt, [https://repositorium.uni-muenster.de/document/miami/265b8283-850a-44a8-845d-f902ff1ca0fa/diss_duchhardt.pdf Thinning Operators and \(\Pi_4\)-Reflection], Munster university doctral thesis, 2008. Accessed 2021-04-16.</ref> and the admissibility for \(\textsf{KP}\omega\) is not equivalent to that for \(\textsf{KP}\). For example, the first admissible ordinal with respect to the admissibility for \(\textsf{KP}\) is \(\omega\), while the first admissible ordinal with respect to the admissibility for \(\textsf{KP} \omega\) is [[Church-Kleene ordinal|\(\omega_1^{\textrm{CK}}\)]] (also the second admissible ordinal with respect to the admissibility for \(\textsf{KP}\)). In this article, we deal with the convention of the admissibility for \(\textsf{KP}\).
 The enumeration function of [[class (set theory)|the class]] of admissible ordinals and limits of admissible ordinals is denoted by \(\alpha \mapsto \omega_{\alpha}^{\textrm{CK}}\){{citation needed}}. For a [[successor function|successor]] ordinal \(\alpha\), \(\omega_\alpha^\text{CK}\) is the \(1+\alpha\)-th admissible ordinal. However, \(\omega_\omega^\text{CK}\) and other ordinals of the form \(\omega_\alpha^\text{CK}\) with [[Limit ordinal|limit]] \(\alpha\) are not necessarily admissible although they are limits of admissibles. In the case that \(\omega_\alpha^\textrm{CK}\) is both admissible and a limit of admissibles, it's also called a '''[[Recursively inaccessible ordinal|recursively inaccessible]]''' ordinal.<ref>D. Madore, [http://www.madore.org/~david/math/ordinal-zoo.pdf A Zoo of Ordinals #2.3] (2017, accessed 2020-11-18)</ref>
-It was believed in this community without proofs 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. In fact, the existence of a system of fundamental sequences such that the property is false is known.<ref>T. Kihara, [http://www.math.mi.i.nagoya-u.ac.jp/~kihara/pdf/misc/omega-1-ck.pdf omega-1-ck.pdf].</ref> Therefore the belief is wrong.
+It was believed in this community without proofs 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. However, there are many ways of defining fundamental sequences (which are necessary to define the FGH), and busy beaver functions with oracles. In fact, a system of fundamental sequences such that the claim is false is known,<ref>T. Kihara, [http://www.math.mi.i.nagoya-u.ac.jp/~kihara/pdf/misc/omega-1-ck.pdf omega-1-ck.pdf].</ref> while a system of fundamental sequences such as the claim is true is ''not'' known.
 ==Properties==
 *If \(\alpha\) is admissible, then there is no \(\Sigma_1\) projection with parameters in \(L_\alpha\) that maps some \(\gamma<\alpha\) to \(\alpha\)<ref>R. Jensen, [https://www.sciencedirect.com/science/article/pii/0003484372900010 The fine structure of the constructible hierarchy] (1972, accessed 2020-11-19) (p.254)</ref>
-*If \(\alpha\) is admissible, then \(L_\alpha\) satisfies a certain \(\Sigma_1\)-separation schema<ref>K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf An introduction to the fine structure of the constructible hierarchy] (1974) (p.17)</ref>.
+*If \(\alpha\) is admissible, then \(L_\alpha\) satisfies a certain \(\Delta_1\)-separation schema<ref>K. Devlin, [https://core.ac.uk/download/pdf/30905237.pdf An introduction to the fine structure of the constructible hierarchy] (1974) (p.19)</ref>.
 ==Extension==
-An extension of the concept of admissibility is \(\Sigma_n\)-admissibility for natural \(n\). An ordinal \(\alpha\) is \(\Sigma_n\)-admissible if \(L_\alpha\) is rudimentary closed and satisfies the [[Kripke–Platek_set_theory#Axiom_Schema_of_.5C.28.5CSigma_0.5C.29-Collection|\(\Sigma_n\)-collection schema]]<ref>E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225 Reflection and partition properties of admissible ordinals] (p.216). Accessed 2021-03-27</ref>. Note that ordinals which are \(\Sigma_n\)-admissible need not satisfy \(\Sigma_n\)-separation, for example the least \(\Sigma_2\)-admissible ordinal doesn't satisfy \(\Sigma_2\)-separation<!--Let this ordinal be α. α satisfies Σ_2-separation, but α-2-stable ordinals below aren't unbounded in α--><ref>M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/HIGH.pdf The Higher Infinite in Proof Theory] (p.20). Accessed 2021-03-27</ref>.
+An extension of the concept of admissibility is \(\Sigma_n\)-admissibility for natural \(n\). An ordinal \(\alpha\) is \(\Sigma_n\)-admissible if \(L_\alpha\) is rudimentary closed and satisfies the [[Kripke–Platek_set_theory#Axiom_Schema_of_.5C.28.5CSigma_0.5C.29-Collection|\(\Sigma_n\)-collection schema]]<ref>E. Kranakis, [https://www.sciencedirect.com/science/article/pii/0003484382900225 Reflection and partition properties of admissible ordinals] (p.216). Accessed 2021-03-27</ref>. Note that ordinals which are \(\Sigma_n\)-admissible need not satisfy \(\Sigma_n\)-separation, for example the least \(\Sigma_2\)-admissible ordinal doesn't satisfy \(\Sigma_2\)-separation<!--Let this ordinal be α. L_α satisfies Σ_2-collection, but α-2-stable ordinals below aren't unbounded in α--><ref>M. Rathjen, [https://www1.maths.leeds.ac.uk/~rathjen/HIGH.pdf The Higher Infinite in Proof Theory] (p.19). Accessed 2021-03-27</ref>.
 == Sources ==
 <references/>