When searching for a "fast growing hierarchy calculator," look for these five features:
Crucial: There is no single FGH. The calculator should let you toggle between: fast growing hierarchy calculator
def fgh(ordinal, n): if n == 0: return 1 # Base case f_a(0) = 1 for a>0 if ordinal == 0: return n + 1 elif isinstance(ordinal, int): # Successor ordinal (e.g., 3 means ω+3) if ordinal == 1: result = n for _ in range(n): result = fgh(0, result) return result else: result = n for _ in range(n): result = fgh(ordinal-1, result) return result else: # Limit ordinal: represented as (base, exponent, coefficient) base, exp, coeff = ordinal # Fundamental sequence for ω^exp * coeff: new_coeff = coeff -1 etc. # (Simplified for illustration) return "Too large to compute directly" When searching for a "fast growing hierarchy calculator,"
To see why a calculator is necessary, try doing f_2(3) by hand: It continues through $\omega+1, \omega \cdot 2, \omega^2,
The hierarchy does not stop at $\omega$. It continues through $\omega+1, \omega \cdot 2, \omega^2, \omega^\omega$, and up to the Feferman–Schütte ordinal ($\Gamma_0$) and far beyond.
The best calculators handle multiple ordinal notations: