User blog:Wythagoras/FGH with transfinite ordinals

Definition (King2218's version)
Base case:

\(f_0(\alpha)=\alpha+1\)

Successor ordinal:

\(f_{\beta+1}(\alpha)=f_{\beta}^\alpha(\alpha)\) for any \(\alpha\) and \(\beta\).

\(f_{\beta}^{\gamma+1}(\alpha)=f_{\beta}(f_{\beta}^{\gamma}(\alpha))\)

\(f_{\beta}^{\gamma+\omega}(\alpha)=\text{lim}\{f_{\beta}^{\gamma}(\alpha),f_{\beta}(f_{\beta}^{\gamma}(\alpha)),f_{\beta}(f_{\beta}(f_{\beta}^{\gamma}(\alpha))),...\}\)

\(f_{\beta}^{\gamma}(\alpha)=\text{lim}_{n\mapsto\omega}f_{\beta}^{\gamma[n]}(\alpha)\)

Limit ordinal:

\(f_{\beta}(\alpha)=f_{\beta[\alpha]}(\alpha)\) for \(\alpha<\omega\)

\(f_{\beta}(\alpha)=\text{lim}_{n\mapsto\omega}f_{\beta[n]}(\alpha)\) for \(\alpha≥\omega\)