The Fast iteration hierarchy is a system similar to the Fast-growing hierarchy. [1]

Definition

f(a){[1]b} = f(f(...)) with b f()

f(a){[b+1]c} = f(a){[b]a[b]a...} with c [b]

f(a){[b]c} = f(a){[ath member of the fundamental sequence of b]c} iff b is a limit ordinal

f(a){[b]c[d]e...} = g(a){[d]e...} where g(a) = f(a){[b]c}

Sources

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