The article on the Fast-growing hierarchy describes a standardized choice for fundamental sequence for all α < ε0.
Even gε0 is only equivalent to f3 and gα only attains the growth of fε0 (the first function that Peano arithmetic cannot prove total in the hierarchy) when α is the Bachmann–Howard ordinal.
[2] Specifically, that there exists an ordinal α such that for all integers n where fα are the functions in the fast-growing hierarchy.
He further showed that the first α this holds for is the ordinal of the theory ID<ω of arbitrary finite iterations of an inductive definition.
[7] The slow-growing hierarchy depends extremely sensitively on the choice of the underlying fundamental sequences.