Fast Growing Hierarchy Calculator High Quality Jun 2026

This is why a is the holy grail for enthusiasts. But what does "high quality" actually mean? This article explores the theory behind FGH, the challenges of implementing it in software, and the features that separate a toy script from a professional-grade ordinal collapsing calculator.

def fgh(alpha, n, limit_ordinal_fundamental=None): """ Compute f_alpha(n) with custom fundamental sequences. Args: alpha: int or callable for limit ordinals returning alpha[n] n: int >= 0 limit_ordinal_fundamental: function(alpha, n) -> alpha_n """ if alpha == 0: return n + 1 if isinstance(alpha, int): # successor result = n for _ in range(n): result = fgh(alpha - 1, result, limit_ordinal_fundamental) return result # limit ordinal if limit_ordinal_fundamental: alpha_n = limit_ordinal_fundamental(alpha, n) return fgh(alpha_n, n, limit_ordinal_fundamental) raise ValueError(f"No fundamental sequence for alpha") fast growing hierarchy calculator high quality

Common choice (Wainer hierarchy):

: This recursion is extremely deep for moderate n (e.g., ( f_\omega+1(3) ) already huge). So high‑quality calculators must: This is why a is the holy grail for enthusiasts

Standard recursion $f_\alpha+1(n) = f_\alpha(f_\alpha(...f_\alpha(n)...))$ is computationally infeasible. to simulate the lower levels of the hierarchy

to simulate the lower levels of the hierarchy. Which of these would be most useful for your research ?