Stability: Difference between revisions
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Stability is a notion and very wide range of types of nonrecursive ordinals, inspired by the weaker notion of reflection. In general, stability is defined via ranks of \(L\) being similar to each other. The weakest type of stability is \((+1)\)-stable, i.e. \(L_\alpha\) being a \(\Sigma_1\)-elementary substructure of \(L_{\alpha+1}\). In general, \(\alpha\) is \(\beta\)-stable, or stable up to \(\beta\), if \(L_\alpha\) is a \(\Sigma_1\)-elementary substructure of \(L_\beta\). The \((+1)\)-stable ordinals are precisely the ordinals which are \(\Pi^1_0\)-reflecting.<ref name="RichterAczel74">Wayne Richter & Peter Aczel, “Inductive Definitions and Reflecting Properties of Admissible Ordinals”, in: Jens Erik Fenstad & Peter G. Hinman (eds.), Generalized Recursion Theory (Oslo, 1972), North-Holland (1974)</ref><sup>Section 6</sup> The \((+2)\)-stable ordinals are \(\Pi^1_0\)-reflecting on the class of \((+1)\)-stable ordinals
This continues on endlessly. \(\Pi^1_1\)-reflection is reached by an \(\alpha\) which is \(\alpha^+\)-stable.<ref name="RichterAczel74" /><sup>Section 6</sup>
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Theorem: Let \(n<\omega\). If \(\alpha\) is \((+n+1)\)-stable, then \(\alpha\) is \(\Pi^1_0\)-reflecting on the class of \((+n)\)-stable ordinals below \(\alpha\).
Proof: Assume \(\alpha\) is (+2)-stable and \(\phi(\vec x)\) is a first-order formula with parameters from \(L_\alpha\) such that \(L_\alpha\vDash\phi(\vec x)\). \(L_{\alpha+2}\) satisfies \(\exists\gamma(\phi^{L_\gamma}(\vec x)\land L_\gamma\prec_{\Sigma_1}L_{\gamma+1})\) with \(\alpha\) as a witness of such a \(\gamma\) (is stability well-behaved in \(L_{\gamma+2}\), a successor stage of \(L\)?), and this is a \(\Sigma_1\) formula, so by \(L_\alpha\prec_{\Sigma_1}L_{\alpha+2}\), \(L_\alpha\) satisfies this as well. Then there is a \(\gamma<\alpha\) such that \(\phi^{L_\gamma}(\vec x)\) and \(\gamma\) is \((+1)\)-stable. QED
Corollary: \(\Pi^1_1\)-reflecting ordinals \(\alpha\) are \(\alpha^+\)-stable, therefore they are \(\alpha+3\)-stable. So each \(\Pi^1_1\)-reflecting ordinal is \(\Pi^1_0\)-reflecting on the \((+2)\)-stable ordinals below it.
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