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. The \((+2)\)-stable ordinals are \(\Pi^1_0\)-reflecting on the class of \((+1)\)-stable ordinals, 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.