Stability

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.^{Section 6} The \((+2)\)-stable ordinals are \(\Pi^1_0\)-reflecting on the class of \((+1)\)-stable ordinals (see section "(+η)-stability") and more, but not yet \(\Pi^1_1\)-reflecting. In general, \((+\eta+1)\)-stable ordinals are significantly greater than \((+\eta)\)-stable ordinals, and then \((\cdot 2)\)-stable ordinals diagonalize over this whole hierarchy. This makes the fine structure of stability useful in OCFs for ordinal analyses of systems beyond Kripke-Platek set theory with full reflection.

This continues on endlessly. \(\Pi^1_1\)-reflection is reached by an \(\alpha\) which is \(\alpha^+\)-stable.^{Section 6}

(+η)-stability

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.