Stable ordinal

Stable ordinals are large ordinals defined using the notion of \(\Sigma_1\)-elementary substructure on levels of constructible universe. ^{(invalid link)}

For a natural number \(n\), let \(M \prec_{\Sigma_n} N\) denote the relation "\(M\) is a \(\Sigma_n\)-elementary substructure of \(N\).

There are special definitions for \(\Sigma_1\)-elementary substructures:

• 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 \(\alpha\).
• 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 \(L_\alpha\prec_{\Sigma_1}L_\gamma\prec_{\Sigma_1}L_{\gamma+\beta}\).^{[citation needed]}
• An ordinal \(\alpha\) is nonprojectible if \(\sup\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}=\alpha\).
• An ordinal \(\alpha\) is stable if \(L_\alpha\prec_{\Sigma_1}L\).

The general definition by Kripke is that for ordinals \(\alpha\) and \(\beta\) with \(\beta>\alpha\), \(\alpha\) is \(\beta\)-stable iff \(L_\alpha\prec_{\Sigma_1}L_\beta\).​ The enumerating function of stable ordinals is continuous.

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") if \(L_\alpha\prec_{\Sigma_n}L_\rho\).

1-\(\rho\)-stability has the following properties: ^{invalid link[citation needed]}

1. If \(\alpha<\beta<\gamma\) and \(L_\alpha\prec_{\Sigma_1}L_\gamma\), then \(L_\alpha\prec_{\Sigma_1}L_\beta\).
2. If \(L_\alpha\prec_{\Sigma_1}L_\beta\) and \(L_\beta\prec_{\Sigma_1}L_\gamma\), then \(L_\alpha\prec_{\Sigma_1}L_\gamma\).
3. If \(\alpha<\beta\) and \(\beta\) is stable, then \(\alpha\) is stable iff \(L_\alpha\prec_{\Sigma_1}L_\beta\).
4. If a set \(A\) is nonempty and \(\forall\alpha\in A(L_\alpha\prec_{\Sigma_1}L_\beta)\), then \(L_{\sup A}\prec_{\Sigma_1}L_\beta\).

For an ordinal \(\alpha\), \(\alpha\) is nonprojectible iff \(L_\alpha\models\text{KP}\omega+\Sigma_1\text{-Sep}\), and "\(\alpha\) is nonprojectible" implies "\(\alpha\) is recursively Mahlo". For a limit ordinal \(\alpha\), \(L_\alpha\models\Sigma_n\text{-Sep}+\Sigma_n\text{-Coll}\) iff \(\forall x\in L_\alpha\exists M\in L_\alpha(x\subseteq M\land M\prec_{\Sigma_n}L_\alpha)\).

Properties of stability:

• (+1)-stable ordinals are exactly \(\Pi_0^1\)-reflecting ordinals.
• \((^+)\)-stable ordinals are exactly \(\Pi_1^1\)-reflecting ordinals.
• 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.
• \(\zeta\), 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.

See also

Ordinals, ordinal analysis, and set theory

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)