= Fundamental sequence of a limit ordinal
{title2=$\lambda[0]<\lambda[1]<\cdots<\lambda,\quad\sup_n\lambda[n]=\lambda$}
= Fundamental sequences of limit ordinals
{synonym}
= Fundamental sequence
{synonym}
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.
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