= Fast-growing hierarchy
{title2=$F_0(n)=n+1,\quad F_{\alpha+1}(n)=F_\alpha^{\,n+1}(n),\quad F_\lambda(n)=F_{\lambda[n]}(n)$}
A fast-growing hierarchy indexes number-theoretic <functions> by an effective system of <ordinal> notations. Successor stages iterate the preceding function, while limit stages diagonalize along a chosen <fundamental sequence of a limit ordinal>. These choices are part of the definition. With the standard <Cantor normal form> sequences below <epsilon zero>, each fixed $F_\alpha$ for $\alpha<\varepsilon_0$ is provably total in <Peano arithmetic>, but the uniform endpoint $F_{\varepsilon_0}$ is not. Every function provably total in <Peano arithmetic> is eventually bounded by some lower level. These classification results are proved in https://epub.ub.uni-muenchen.de/3843/1/3843.pdf .
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