[0] Pre-alpha stage: Primitive recursive function level ~ 0 to the limit of Hyper-E regiment level, plus micronyms (FGH ordinal level 0 ~ ω).
[1] Alpha stage: Recursive levels eventually dominate all primitive recursive function levels but provable within Peano arithmetic (Extended Hyper-E level to the limit of Cascading-E level ~ FGH ordinal level ω ~ ε0). The fast-growing hierarchy level in comparison is given by Wainer hierarchy.
[2] Beta stage: Higher recursive levels eventually dominate all hyperrecursive functions and the totality of these functions is not provable in Peano arithmetic (Extended Cascading-E level to the limit of Collapsing-E level ~ FGH ordinal level ε0 ~ BHO).
[3] Gamma stage: Higher recursive levels eventually dominate all functions provably recursive in Kripke-Platek set theory with the axiom of infinity (KP) (Extended Collapsing-E level to the limit of Hyper-Extended Collapsing-E level ~ FGH ordinal level BHO ~ EBO).
[4] Delta stage: Higher recursive levels eventually dominate all functions provably recursive in П1^1-TR0 (beyond the Hyper-Extended Collapsing-E level ~ FGH ordinal level >EBO).