Stable

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$f$-Stable Ordinals

An ordinal $\alpha$ is $f$-stable for a function $f$ such that $\alpha\leq f(\alpha)$ iff $L_{\alpha}\preceq_{1}L_{f(\alpha)}$. For example:

The smallest $\Pi_{0}^1$-Reflective ordinal is (+1)-stable.

$\beta$-Stable Ordinals

An ordinal $\alpha$ is $\beta$-stable for an ordinal $\beta$ such that $\alpha\leq\beta$ iff $L_{\alpha}\preceq_{1}L_{\beta}$.

An ordinal $\alpha$ is stable iff $L_{\alpha}\preceq_{1}L_{\omega_{1}}$. The smallest stable ordinal is also the smallest $\Sigma_{2}^1$-reflecting ordinal.

References


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