Stable ordinals are large ordinals defined using the notion of \(\Sigma_1\)-elementary substructure on levels of constructible universe.[1][2][footnote 1]
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 \(L_\alpha \prec_{\Sigma_1} L\).[2] The following are various weakenings of the notion of stability:
- An ordinal \(\alpha\) is \((+\beta)\)-stable if \(L_\alpha\prec_{\Sigma_1}L_{\alpha+\beta}\).[citation needed]
- An ordinal \(\alpha\) is \((^+)\)-stable if \(L_\alpha\prec_{\Sigma_1}L_\beta\) where \(\beta\) is next admissible ordinal after \(\alpha\).[2]
- 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 \(\alpha\).[2]
- An ordinal \(\alpha\) is inaccessibly-stable if \(L_\alpha\prec_{\Sigma_1}L_\beta\) where \(\beta\) is next recursively inaccessible ordinal after \(\alpha\).[2]
- An ordinal \(\alpha\) is Mahlo-stable if \(L_\alpha\prec_{\Sigma_1}L_\beta\) where \(\beta\) is next recursively Mahlo ordinal after \(\alpha\).[2]
- An ordinal \(\alpha\) is doubly \((+\beta)\)-stable if there exists ordinal \(\gamma\) such that \(L_\alpha\prec_{\Sigma_1}L_\gamma\prec_{\Sigma_1}L_{\gamma+\beta}\).[citation needed]
- An ordinal \(\alpha\) is nonprojectible if \(\sup\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}=\alpha\).[2] Nonprojectible \(\alpha\) are also \(\Pi_2\)-reflecting on \(\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}\). [3][footnote 2]
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.[4][5]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"[6]) if \(L_\alpha\prec_{\Sigma_n}L_\rho\)[7]p.19.
1-\(\rho\)-stability has the following properties:[5]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 \(L_\alpha\models\text{KP}\omega+\Sigma_1\text{-Sep}\)[7]p.19, and "\(\alpha\) is nonprojectible" implies "\(\alpha\) is recursively Mahlo"[5]p.188. For a limit ordinal \(\alpha\), \(L_\alpha\models\Sigma_n\text{-Sep}+\Sigma_n\text{-Coll}\) iff \(\forall x\in L_\alpha\exists M\in L_\alpha(x\subseteq M\land M\prec_{\Sigma_n}L_\alpha)\).[7]p.19
Properties of stability:
- (+1)-stable ordinals are exactly \(\Pi_0^1\)-reflecting ordinals.[8]p.44
- \((^+)\)-stable ordinals are exactly \(\Pi_1^1\)-reflecting ordinals.[8]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.[9]
- \(\zeta\), 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.[10]
Footnotes[]
- ↑ 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.
- ↑ The proof of interest ((ii)→(iii)) does not require \(\Sigma_2\textrm{-cof}(\alpha)>\omega\) or \(J_\alpha\vDash\lnot\textrm{Powerset axiom}\).
Sources[]
- ↑ Cantor's attic. Stable Update 2020-10-19. Retrieved 2023-09-15.
- ↑ 2.0 2.1 2.2 2.3 2.4 2.5 2.6 David A. Madore, A zoo of ordinals.
- ↑ E. Kranakis (1982) Reflection and partition properties of admissible ordinals Annals of Mathematical Logic 22:3. pp.213-242.
- ↑ W. Marek and M. Srebrny (1974) Gaps in the Constructible Universe Annals of Mathematical Logic 6:3–4. pp.359-394 (p.382)
- ↑ 5.0 5.1 5.2 J. Barwise (2017) "Admissible Sets and Structures", Cambridge University Press.
- ↑ 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
- ↑ 7.0 7.1 7.2 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.
- ↑ 8.0 8.1 Richter & Aczel (1974) Inductive Definitions and Reflecting Properties of Admissible Ordinals Studies in Logic and the Foundations of Mathematics 79. pp.301-381 Preprint.
- ↑ 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
- ↑ Ansten Morch Klev (2007) "Extending Kleene’s O Using Infinite Time Turing Machines" MSc Thesis. Universiteit van Amsterdam.
See also[]
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)