Stable ordinal

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

We work in \(\textrm{ZFC}\) set theory, and abbreviate "definable class" to "class" for simplicity. Let \(n\) be a natural number. A class \(M\) is a \(\Sigma_n\)-elementary substructure of a class \(N\) if it is a substructure of \(N\) and for every \(\Sigma_n\) formula \(\phi(\vec{p})\), \(M\models^n \phi(\vec{p})\leftrightarrow N\models^n \phi(\vec{p})\), denoted \(M\prec_{\Sigma_n}N\), where \(\models^n\) denotes Levy's truth predicate on classes and \(\Sigma_n\) formulae. Since we are only considering definable classes, the truth predicate is actually formalisable in \(\textrm{ZFC}\) set theory.

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

There are special definitions for \(\Sigma_1\)-elementary substructures: The enumerating function of stable ordinals is continuous.
 * An ordinal \(\alpha\) is \((+\beta)\)-stable if \(L_\alpha\prec_{\Sigma_1}L_{\alpha+\beta}\).
 * 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 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 nonprojectable if \(\sup\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}=\alpha\).
 * An ordinal \(\alpha\) is stable if \(L_\alpha\prec_{\Sigma_1}L\).

Suppose \(n\ge1\). An ordinal \(\alpha\) is \(n\)-stable if \(L_\alpha\prec_{\Sigma_n}L\). For \(\rho>\alpha\), an ordinal \(\alpha\) is \((\rho,n)\)-stable if \(L_\alpha\prec_{\Sigma_n}L_\rho\).

\((\rho,1)\)-stability has the following properties: invalid link For an ordinal \(\alpha\), \(\alpha\) is nonprojectable iff \(L_\alpha\models\text{KP}\omega+\Sigma_1\text{-Sep}\). For a limit ordinal \(\alpha\), \(L_\alpha\models\Sigma_n\text{-Sep}+\Sigma_n\text{-Coll}\) iff \(\forall x\in L_\alpha\exists M\in L_\alpha(x\subseteq M\land M\prec_{\Sigma_n}L_\alpha)\).
 * 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 \(L_{\sup A}\prec_{\Sigma_1}L_\beta\).

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 \(\alpha\) that is \((\rho,2)\)-stable for some \(\rho\); and \(\Sigma\), the supremum of all accidentally writable ordinals, is the least ordinal \(\rho\) such that \(\zeta\) is \((\rho,2)\)-stable.