Stable ordinal

Stable ordinals are large ordinals defined using the notion of \(\Sigma_1\)-elementary substructure on levels of constructible universe.Cantor's attic. Stable Update 2020-10-19. Retrieved 2023-09-15.David A. Madore, A zoo of ordinals.Here, \(\Sigma_1\) denotes Levy's hierarchy. (reference 4) On the other hand, Madore's zoo refers to \(\Pi\), \(\Sigma\), and \(\Delta\) as the hierarchy for arithmetic in other parts. If it is actually intended, then the statement on the reflection property of a \((+1)\)-stable ordinal conflicts the definition based on \(\Sigma_1\)-elementary substructure. Therefore it might be a mistake, or simply an abuse of notation.
For a natural number \(n\), let \(M \prec_{\Sigma_n} N\) denote the relation "\(M\) is a \(\Sigma_n\)-elementary substructure of \(N\). Then an ordinal \(\alpha\) is stable if \(L_\alpha \prec_{\Sigma_1} L\). The following are various weakenings of the notion of stability:
*An ordinal \(\alpha\) is \(\beta\)-stable if \(L_\alpha\prec_{\Sigma_1}L_{\beta}\).
*An ordinal \(\alpha\) is \((^+)\)-stable if \(L_\alpha\prec_{\Sigma_1}L_\beta\) where \(\beta\) is next admissible ordinal after \(\alpha\).
*An ordinal \(\alpha\) is \((^{++})\)-stable if \(L_\alpha\prec_{\Sigma_1}L_\beta\) where \(\beta\) is next admissible ordinal after the next admissible ordinal after \(\alpha\).
*An ordinal \(\alpha\) is inaccessibly-stable if \(L_\alpha\prec_{\Sigma_1}L_\beta\) where \(\beta\) is next recursively inaccessible ordinal after \(\alpha\).
*An ordinal \(\alpha\) is Mahlo-stable if \(L_\alpha\prec_{\Sigma_1}L_\beta\) where \(\beta\) is next recursively Mahlo ordinal after \(\alpha\).
*An ordinal \(\alpha\) is doubly \((+\beta)\)-stable if there exists ordinal \(\gamma\) such that \(L_\alpha\prec_{\Sigma_1}L_\gamma\prec_{\Sigma_1}L_{\gamma+\beta}\).
*An ordinal \(\alpha\) is nonprojectible if \(\sup\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}=\alpha\). Nonprojectible \(\alpha\) are also \(\Pi_2\)-reflecting on \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\). E. Kranakis (1982) Reflection and partition properties of admissible ordinals Annals of Mathematical Logic 22:3. pp.213-242.The proof of interest ((ii)→(iii)) does not require \(\Sigma_2\textrm{-cof}(\alpha)>\omega\) or \(J_\alpha\vDash\lnot\textrm{Powerset axiom}\).
The general definition by Kripke is that for ordinals \(\alpha\) and \(\beta\) with \(\beta>\alpha\), \(\alpha\) is \(\beta\)-stable iff \(L_\alpha\prec_{\Sigma_1}L_\beta\).[​3]​​
The enumerating function of stable ordinals is continuous.W. Marek and M. Srebrny (1974) Gaps in the Constructible Universe Annals of Mathematical Logic 6:3–4. pp.359-394 (p.382)p.178
Suppose \(n\ge 1\). An ordinal \(\alpha\) is \(n\)-stable if \(L_\alpha\prec_{\Sigma_n}L\). For \(\rho>\alpha\), an ordinal \(\alpha\) is \(n\)-\(\rho\)-stable (also called "\((\rho,n)\)-stable"T. Arai (2012) A sneak preview of proof theory of ordinals Annals of the Japan Association for Philosophy of Science vol. 20, pp. 29-47. Preprint p.13) if \(L_\alpha\prec_{\Sigma_n}L_\rho\)M. Rathjen, The Higher Infinite in Proof Theory in J.A. Makowsky, V. Harnik (eds.), Proceedings of the Annual European Summer Meeting of the Association of Symbolic Logic, Haifa, Israel, August 9–18, 1995. pp.275-304.p.19.
1-\(\rho\)-stability has the following properties:J. Barwise (2017) "Admissible Sets and Structures", Cambridge University Press.p.179
#If \(\alpha<\beta<\gamma\) and \(L_\alpha\prec_{\Sigma_1}L_\gamma\), then \(L_\alpha\prec_{\Sigma_1}L_\beta\).
#If \(L_\alpha\prec_{\Sigma_1}L_\beta\) and \(L_\beta\prec_{\Sigma_1}L_\gamma\), then \(L_\alpha\prec_{\Sigma_1}L_\gamma\).
#If \(\alpha<\beta\) and \(\beta\) is stable, then \(\alpha\) is stable iff \(L_\alpha\prec_{\Sigma_1}L_\beta\).
#If a set \(A\) is nonempty and \(\forall\alpha\in A(L_\alpha\preceq_{\Sigma_1}L_\beta)\), then \(L_{\sup A}\prec_{\Sigma_1}L_\beta\).
For an ordinal \(\alpha\), \(\alpha\) is nonprojectible iff \(L_\alpha\models\text{KP}\omega+\Sigma_1\text{-Sep}\)p.19, and "\(\alpha\) is nonprojectible" implies "\(\alpha\) is recursively Mahlo"p.188. For a limit ordinal \(\alpha\), \(L_\alpha\models\Sigma_n\text{-Sep}+\Sigma_n\text{-Coll}\) iff \(\forall x\in L_\alpha\exists M\in L_\alpha(x\subseteq M\land M\prec_{\Sigma_n}L_\alpha)\).p.19
Properties of stability:
* (+1)-stable ordinals are exactly \(\Pi_0^1\)-reflecting ordinals.Richter & Aczel (1974) Inductive Definitions and Reflecting Properties of Admissible Ordinals Studies in Logic and the Foundations of Mathematics 79. pp.301-381 Preprint.</sup>p.44</sup>
* \((^+)\)-stable ordinals are exactly \(\Pi_1^1\)-reflecting ordinals.p.16
* The least \(\Sigma_1^1\)-reflecting ordinal is greater than the least \(\Pi_1^1\)-reflecting ordinal, but every \((^{++})\)-stable ordinal is \(\Sigma_1^1\)-reflecting.
* Stable ordinals are exactly \(\Sigma_2^1\)-reflecting ordinals, while the least of them is less than the least \(\Pi_2^1\)-reflecting ordinal.W. Richter (2006) The Least \(\Sigma^1_2\) and \(\Pi^1_2\) Reflecting Ordinals, ISILC Logic Conference, Lecture Notes in Mathematics, vol 499. Springer, Berlin, Heidelberg. ISBN 3-540-07534-8, 568-578
* \(\zeta\), the supremum of all eventually writable ordinals, is the least ordinal that is 2-\(\rho\)-stable for some \(\rho\); and \(\Sigma\), the supremum of all accidentally writable ordinals, is the least ordinal \(\rho\) such that \(\zeta\) is 2-\(\rho\)-stable.Ansten Morch Klev (2007) "Extending Kleene’s O Using Infinite Time Turing Machines" MSc Thesis. Universiteit van Amsterdam.
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== See also ==
ja:安定順序数
Category:Transfinite ordinals
Category:Set theory