User:Augigogigi/Mahlo Notation

• \( \alpha \vartriangleleft_{0} A \) is true iff for all functions \( f : \alpha \rightarrow \alpha \), there exists \( \kappa \in A \) that is closed under \( f \)
• \( [0]M = \{ \alpha : \alpha \vartriangleleft Ord \} \)
• \( [S,\alpha + 1]\Xi_{0} = \{ \beta : \beta \vartriangleleft [\alpha]\Xi_{0} \} \)
• \( [S,\alpha]\Xi_{0} = \bigcap_{\beta<\alpha} [\beta]\Xi_{0} \) iff \( \alpha \in Lim \)
• \( [1,0,S]\Xi_{0} = \{ \alpha : \alpha = \bigcap_{\beta<\alpha} [\beta,S]\Xi_{0} \} \)
• \( A(\alpha) = \text{enum}(A) \)
• \( r(A,[\alpha]\Xi_{0}(S,\beta + 1,0)) = \{ \gamma : \gamma = [\alpha]\Xi_{0}(S,\beta,\gamma) \land \gamma \in A \} \)