 Fundamental sequence of a limit ordinal

For a countable limit ordinal, a fundamental sequence is an increasing cofinal sequence of smaller ordinals. Effective choices allow a fast-growing hierarchy to turn limit recursion into recursion at a finite input. Below epsilon zero, write the final term as $\gamma +{\omega }^{\beta }$. For successor $\beta =\delta +1$, take $\gamma +{\omega }^{\delta }(n+1)$; for nonzero limit $\beta$, take $\gamma +{\omega }^{\beta [n]}$. Earlier terms in the Cantor normal form remain fixed.