Stable ordinal

Stable ordinals are large ordinals. The notion has two distinct definitions: the definition in Madore's zoo, and the definition in Cartor's attic. Although this community usually implicitly refers to the definition in Cantor's attic, readers should be very careful that results in mathematics usually refer to the definition in Madore's zoo. (This article referred to those results as theorems on the definition in Cantor's attic, but they actually refer to the definition in Madore's zoo. Namely, the applications were wrong.)

Definition in Cantor's attic
Stable ordinals are defined using the notion of \(\Sigma_1\)-elementary substructure on levels of constructible universe. 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.

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

There are special definitions for \(\Sigma_1\)-elementary substructures: (invalid link) \((\rho,1)\)-stability has the following properties: invalid link
 * A countable ordinal \(\alpha\) is \((+\beta)\)-stable if \(L_\alpha\prec_{\Sigma_1}L_{\alpha+\beta}\).
 * A countable ordinal \(\alpha\) is \((^+)\)-stable if \(L_\alpha\prec_{\Sigma_1}L_\beta\) where \(\beta\) is next admissible ordinal after \(\alpha\).
 * A countable ordinal \(\alpha\) is \((^{++})\)-stable if \(L_\alpha\prec_{\Sigma_1}L_\beta\) where \(\beta\) is next admissible ordinal after the admissible ordinal after \(\alpha\).
 * A countable ordinal \(\alpha\) is inaccessibly-stable if \(L_\alpha\prec_{\Sigma_1}L_\beta\) where \(\beta\) is next recursively inaccessible ordinal after \(\alpha\).
 * A countable ordinal \(\alpha\) is Mahlo-stable if \(L_\alpha\prec_{\Sigma_1}L_\beta\) where \(\beta\) is next recursively Mahlo ordinal after \(\alpha\).
 * A countable ordinal \(\alpha\) is doubly \((+\beta)\)-stable if there exists ordinal \(\gamma\) such that \(L_\alpha\prec_{\Sigma_1}L_\gamma\prec_{\Sigma_1}L_{\gamma+\beta}\).
 * A countable ordinal \(\alpha\) is nonprojectable if \(\sup\{\beta<\alpha|L_\beta\prec_{\Sigma_1}L_\alpha\}=\alpha\).
 * A countable ordinal \(\alpha\) is stable if \(L_\alpha\prec_{\Sigma_1}L\).
 * 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 \(L_{\sup A}\prec_{\Sigma_1}L_\beta\).

Definition in Madore's zoo
Stable ordinals are defined using the reflection property for arithmetic formulae in a completely similar way above.

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.

For a countable 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 \(\forall x\in L_\alpha\exists M\in L_\alpha(x\subseteq M\land M\prec_{\Sigma_n}L_\alpha)\).