Stability: Difference between revisions
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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 \(\alpha\). |
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 \(\alpha\). |
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Proof: Assume \(\alpha\) is (+2)-stable and \(\phi(\vec x)\) is a first-order formula with parameters from \(L_\alpha\) such that \(L_\alpha\vDash\phi(\vec x)\). |
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 |
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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. |
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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